Trigonometry

1. Revision



ID is: 4369 Seed is: 6886

CAST diagram: problem solving to find coordinates

Adapted from DBE Nov 2015 Grade 11, P2, Q5.1
Maths formulas

In the diagram below, D(24;7) is a point such that OD=25 and EO^D=θ, where θ is an obtuse angle.

NOTE: The following diagram is not drawn to scale.

F is a point on OD such that OF=2. Determine the coordinates of F without using a calculator.

INSTRUCTION: Type brackets around the x- and y-coordinates, and type a semicolon in between, like this: (x ; y)
Answer: Coordinates of F:
coordinate

ID is: 4369 Seed is: 5728

CAST diagram: problem solving to find coordinates

Adapted from DBE Nov 2015 Grade 11, P2, Q5.1
Maths formulas

In the diagram below, P(20;21) is a point such that OP=29 and RO^P=α, where α is a reflex angle.

NOTE: The following diagram is not drawn to scale.

T is a point on OP such that OT=3. Determine the coordinates of T without using a calculator.

INSTRUCTION: Type brackets around the x- and y-coordinates, and type a semicolon in between, like this: (x ; y)
Answer: Coordinates of T:
coordinate

2. Trigonometric identities



ID is: 3465 Seed is: 6475

Where does the square identity come from?

The square identity is

sin2θ+cos2θ=1

The two questions below are about the origin of this identity.

  1. Which of these choices is the origin of the square identity?

    Equation A cosθ=adjacenthypotenuse
    Equation B tanθ=sinθcosθ
    Equation C a2+b2=c2
    Equation D tanθ=m
    Answer:

    The origin of the square identity is .

  2. The table below shows a proof of the square identity. The proof consists of 6 steps and reasons. However, the proof has one missing step and one missing reason. Choose the correct step and reason to complete the proof.

    Answer:
    Steps Reasons
    Given
    x2+y2=r2 Theorem of Pythagoras
    x2r2+y2r2=1 Multiply by 1r2
    cosθ=xr on Cartesian plane
    cos2θ+sin2θ=1 sinθ=yr on Cartesian plane
    sin2θ+cos2θ=1

ID is: 3465 Seed is: 2194

Where does the square identity come from?

The square identity is

sin2θ+cos2θ=1

The two questions below are about the origin of this identity.

  1. Which of these choices is the origin of the square identity?

    Equation A a2+b2=c2
    Equation B cosθ=adjacenthypotenuse
    Equation C tanθ=m
    Equation D tanθ=oppositeadjacent
    Answer:

    The origin of the square identity is .

  2. The table below shows a proof of the square identity. The proof consists of 6 steps and reasons. However, the proof has one missing step and one missing reason. Choose the correct step and reason to complete the proof.

    Answer:
    Steps Reasons
    Given
    x2+y2=r2 Theorem of Pythagoras
    x2r2+y2r2=1
    cos2θ+y2r2=1 cosθ=xr on Cartesian plane
    sinθ=yr on Cartesian plane
    sin2θ+cos2θ=1 Commutativity of addition


ID is: 3577 Seed is: 4362

Using the square identity to simplify an expression

Here is an expression which includes the sine and cosine functions.

sin2x+cos2x+6

Use the square identity to simplify this expression.

Answer:

The simplified answer is: .

numeric

ID is: 3577 Seed is: 2269

Using the square identity to simplify an expression

Here is an expression which includes the sine and cosine functions.

4+sin2x+cos2x

Use the square identity to simplify this expression.

Answer:

The simplified answer is: .

numeric


ID is: 3576 Seed is: 2914

Simplifying an identity using tanA=sinAcosA

Use the quotient identity to simplify the following expression:

5cosBtanB+5
INSTRUCTION: If you need to type a trigonometric expression, put brackets around the angle. For example: sin(B) or cos(B).
Answer:

The simplified answer is: .

expression

ID is: 3576 Seed is: 3349

Simplifying an identity using tanA=sinAcosA

Use the quotient identity to simplify the following expression:

5cosBtanB5
INSTRUCTION: If you need to type a trigonometric expression, put brackets around the angle. For example: sin(B) or cos(B).
Answer:

The simplified answer is: .

expression


ID is: 3609 Seed is: 3970

Proofs: trigonometric identities

Consider the following equation:

4=4(sinψ+cosψ)(tanψcosψcosψ)+8cos2ψ

Prove this equation is an identity. (Select the correct steps and reasons to complete the proof below.)

Answer:
Steps Reasons
4(sinψ+cosψ)(tanψcosψcosψ)+8cos2ψ Given
Quotient identity substitution
4(sinψ+cosψ)(sinψcosψ)+8cos2ψ
4(sin2ψcos2ψ)+8cos2ψ Multiply the brackets
4sin2ψ4cos2ψ+8cos2ψ Distribute the 4
4sin2ψ+4cos2ψ
Factorisation
4 Evaluate

ID is: 3609 Seed is: 6340

Proofs: trigonometric identities

Consider the following equation:

4(sinλ+cosλ)(tanλcosλcosλ)+8cos2λ=4

Prove this equation is an identity. (Select the correct steps and reasons to complete the proof below.)

Answer:
Steps Reasons
4(sinλ+cosλ)(tanλcosλcosλ)+8cos2λ Given
Quotient identity substitution
4(sinλ+cosλ)(sinλcosλ)+8cos2λ
4(sin2λcos2λ)+8cos2λ Multiply the brackets
Distribute the 4
4sin2λ+4cos2λ
Factorisation
4(1)
4 Evaluate


ID is: 3582 Seed is: 3497

Identity or not?

In maths, an identity is a specific type of equation. For each equation below, state if it is an identity or not.

Answer:
Equation Identity or not
(2x)(2x3)=4x4
x22x3=0
5x+4x=9x

ID is: 3582 Seed is: 2210

Identity or not?

In maths, an identity is a specific type of equation. For each equation below, state if it is an identity or not.

Answer:
Equation Identity or not
(4x)(5x3)=20x4
2x+2=x+3
4(x+1)=4x+4


ID is: 3606 Seed is: 2643

Proofs: trigonometric identities

Answer the two questions below about this identity:

8cos2A41sinA=41+sinA
  1. The table below shows a proof for the identity. But there are three steps and three reasons missing from the proof. Select the correct choices from the lists to complete the proof.

    Answer:
    Steps Reasons
    8cos2A41sinA Given
    Multiply by (1+sinA)(1+sinA)=1
    8cos2A4(1+sinA)(1sinA)(1+sinA)
    8cos2A4(1+sinA)(1sin2A) Expand brackets
    84(1+sinA)(1sin2A) Add fractions with equal denominators
    844sinA(1sin2A) Distribute
    44sinA(1+sinA)(1sinA) Factorise difference of two squares
    4(1sinA)(1+sinA)(1sinA) Factorise out 4
    41+sinA Cancel (1sinA) factors
  2. For which value(s) of A in the interval 180°A360° is the identity in Question 1 undefined? The identity is:

    8cos2A41sinA=41+sinA
    Answer:

    The identity is undefined when A= .


ID is: 3606 Seed is: 8111

Proofs: trigonometric identities

Answer the two questions below about this identity:

12sin2B61cosB=61+cosB
  1. The table below shows a proof for the identity. But there are three steps and three reasons missing from the proof. Select the correct choices from the lists to complete the proof.

    Answer:
    Steps Reasons
    12sin2B61cosB Given
    12sin2B6(1cosB)(1+cosB)(1+cosB)
    12sin2B6(1+cosB)(1cosB)(1+cosB) Multiply fractions
    12(1cos2B)6(1+cosB)(1cos2B) Square identity substitution
    126(1+cosB)(1cos2B) Add fractions with equal denominators
    1266cosB(1cos2B) Distribute
    Collect constant terms
    66cosB(1+cosB)(1cosB) Factorise difference of two squares
    61+cosB Cancel (1cosB) factors
  2. For which value(s) of B in the interval 180°B360° is the identity in Question 1 undefined? The identity is:

    12sin2B61cosB=61+cosB
    Answer:

    The identity is undefined when B= .



ID is: 4360 Seed is: 1872

Evaluate an expression using trig identities

Adapted from DBE Nov 2015 Grade 11, P2, Q5.3
Maths formulas

Evaluate the following expression:

5cosθsinθ(1+tan2θ)tanθ
Answer:

numeric

ID is: 4360 Seed is: 2274

Evaluate an expression using trig identities

Adapted from DBE Nov 2015 Grade 11, P2, Q5.3
Maths formulas

Determine the value of the following expression:

sin3θ+sinθcos2θ+2cosθtanθsinθ
Answer:

numeric


ID is: 3588 Seed is: 1568

Asymptotes of the tangent function

The tangent function, f(β)=tanβ, has vertical asymptotes. Answer the following questions about these asymptotes.

  1. In the interval 360°β360°, at what values of β are the asymptotes located?

    Answer:

    The asymptotes are at β= .

  2. How are the asymptotes related to the quotient identity?

    tanβ=sinβcosβ
    Answer:

    The asymptotes of tanβ .


ID is: 3588 Seed is: 4493

Asymptotes of the tangent function

The tangent function, f(β)=tanβ, has vertical asymptotes. Answer the following questions about these asymptotes.

  1. In the interval 360°β360°, at what values of β are the asymptotes located?

    Answer:

    The asymptotes are at β= .

  2. How are the asymptotes related to the quotient identity?

    tanβ=sinβcosβ
    Answer:

    The asymptotes of tanβ .



ID is: 3650 Seed is: 7704

Undefined cases

Answer the two questions below about this identity:

cosθsinθtanθ=cos2θ
  1. For what values of θ in the interval [0°;360°] is the identity undefined? If the identity is never undefined, select the identity is never undefined.

    Answer:

    In the interval [0°;360°], the identity is undefined for θ .

  2. The equation in Question 1 is an identity. Select the best choice below to complete the sentence about identities. The variable θ represents an angle in the identity.

    An identity is an equation that...
    A is true for all values of θ for which both sides are defined.
    B is false when there is an undefined value
    C is true for all values of θ which are inside the interval.
    D is true unless it has no solution.
    Answer: The correct choice is .

ID is: 3650 Seed is: 3190

Undefined cases

Answer the two questions below about this identity:

sinθcosθtanθ=cos2θ
  1. For what values of θ in the interval [0°;360°] is the identity undefined? If the identity is never undefined, select the identity is never undefined.

    Answer:

    In the interval [0°;360°], the identity is undefined for θ .

  2. The equation in Question 1 is an identity. Select the best choice below to complete the sentence about identities. The variable θ represents an angle in the identity.

    An identity is an equation that...
    A is true unless it has no solution.
    B is true for all values of θ which are inside the interval.
    C is true for all values of θ for which both sides are defined.
    D is false for values of θ which are negative.
    Answer: The correct choice is .


ID is: 3612 Seed is: 5411

Simplifying trigonometric identities

  1. Simplify the following expression as much as possible:

    sinA+(3+cosA)tanA
    INSTRUCTION: If you need to type a trigonometric expression, put brackets around the angle. For example: sin(A) or cos(A).
    Answer:

    The simplified expression is .

    expression
  2. What is an identity? Choose the correct words to complete the definition below.

    Answer:

    An identity is which as long as both sides are defined.


ID is: 3612 Seed is: 5043

Simplifying trigonometric identities

  1. Simplify the following expression as much as possible:

    2+tanAcosA
    INSTRUCTION: If you need to type a trigonometric expression, put brackets around the angle. For example: sin(A) or cos(A).
    Answer:

    The simplified expression is .

    expression
  2. What is an identity? Choose the correct words to complete the definition below.

    Answer:

    An identity is which as long as both sides are defined.



ID is: 3578 Seed is: 837

Simplifying an identity using sin2A+cos2A=1

Simplify the following expression. Use the square identity.

3cos2B+3sin2B+2
Answer:

The simplified answer is: .

numeric

ID is: 3578 Seed is: 4881

Simplifying an identity using sin2A+cos2A=1

Simplify the following expression. Use the square identity.

5(sinA+cosA)(sinAcosA)+10cos2A+2
Answer:

The simplified answer is: .

numeric


ID is: 3610 Seed is: 1188

Simplifying trigonometric identities

Consider the following expression and answer the two questions below:

6+cos2x+sin2x
  1. If we must simplify this expression, which of these is most likely to be the better choice?

    Quotient identity:
    tanx=sinxcosx
    Square identity:
    sin2x+cos2x=1
    Answer:

    The better choice is the .

  2. Now simplify the expression as much as possible:

    6+cos2x+sin2x
    INSTRUCTION: If you need to type a trigonometric expression, put brackets around the angle. For example: sin(x) or cos(x).
    Answer:

    The simplified expression is .

    numeric

ID is: 3610 Seed is: 9079

Simplifying trigonometric identities

Consider the following expression and answer the two questions below:

4cosxtanx
  1. If we must simplify this expression, which of these is most likely to be the better choice?

    Quotient identity:
    tanx=sinxcosx
    Square identity:
    sin2x+cos2x=1
    Answer:

    The better choice is the .

  2. Now simplify the expression as much as possible:

    4cosxtanx
    INSTRUCTION: If you need to type a trigonometric expression, put brackets around the angle. For example: sin(x) or cos(x).
    Answer:

    The simplified expression is .

    expression


ID is: 1442 Seed is: 8451

Different forms of the quotient identity

The quotient identity is:

tanθ=sinθcosθ

This equation can be arranged in different ways. If it is arranged starting as shown, which of the expressions in the list belongs on the right side:

cosθ=??
Answer:

ID is: 1442 Seed is: 3254

Different forms of the quotient identity

The quotient identity is:

tanα=sinαcosα

This equation can be arranged in different ways. If it is arranged starting as shown, which of the expressions in the list belongs on the right side:

cosαsinα=??
Answer:


ID is: 3608 Seed is: 2957

Building a proof

Answer the two questions below about this identity:

35=3(1cosθ)(1+cosθ)5sin2θ
  1. To prove this identity, you must start with one side of the equation and change it until you get the other side of the equation. In this case, which side of the identity is the better choice to start with?

    Answer:

    We should start with .

  2. The table below shows the proof for this identity. It is 4 steps long. However, the proof is not done yet: there are 2 steps and 2 reasons missing. Select the correct steps and reasons from the choices available.

    Answer:
    Steps Reasons
    3(1cosθ)(1+cosθ)5sin2θ
    3(1cos2θ)5sin2θ
    Square identity substitution
    Cancel sinθ factors

ID is: 3608 Seed is: 6285

Building a proof

Answer the two questions below about this identity:

tanθtanθsin2θcos2θ=tanθ
  1. To prove this identity, you must start with one side of the equation and change it until you get the other side of the equation. In this case, which side of the identity is the better choice to start with?

    Answer:

    We should start with .

  2. The table below shows the proof for this identity. It is 4 steps long. However, the proof is not done yet: there are 2 steps and 2 reasons missing. Select the correct steps and reasons from the choices available.

    Answer:
    Steps Reasons
    tanθtanθsin2θcos2θ
    tanθ(1sin2θ)cos2θ
    Square identity substitution
    Cancel cosine factors


ID is: 3648 Seed is: 4514

When is an identity undefined?

  1. It is possible for an identity to be undefined. There are two situations that can make an identity undefined. What are these two situations?

    Answer:

    An identity is undefined if or if .

  2. Is the following identity ever undefined?

    tanA(1cosA)+sinA=tanA
    Answer:

ID is: 3648 Seed is: 4743

When is an identity undefined?

  1. It is possible for an identity to be undefined. There are two situations that can make an identity undefined. What are these two situations?

    Answer:

    An identity is undefined if or if .

  2. Is the following identity ever undefined?

    5cos2A+5tanAcosAsinA=5
    Answer:


ID is: 3649 Seed is: 7656

Finding values for which an identity is undefined

The following identity is undefined for some values of A.

tanA(3+cosA)sinA=3tanA

In the interval 360°A360°, what are the values of A which make the identity undefined?

Answer:

For 360°A360°, the identity is undefined for A .


ID is: 3649 Seed is: 2862

Finding values for which an identity is undefined

The following identity is undefined for some values of B.

sinBtanB(cosB2)=2tanB

In the interval 360°B360°, what are the values of B which make the identity undefined?

Answer:

For 360°B360°, the identity is undefined for B .



ID is: 1443 Seed is: 2662

Different forms of the square (Pythagorean) identity.

The square identity is:

sin2A+cos2A=1

This equation can be rearranged. If the equation is arranged starting as shown here, which expression belongs on the right side of the equation?

sin2A=??

Answer Choices:

1 (1cosA)(1+cosA)
2 (1+cosA)(1+cosA)
3 (1sinA)(1+sinA)
4 1sin2A
Answer: The correct expression is: .

ID is: 1443 Seed is: 6785

Different forms of the square (Pythagorean) identity.

The square identity is:

sin2β+cos2β=1

This equation can be rearranged. If the equation is arranged starting as shown here, which expression belongs on the right side of the equation?

cos2β=??

Answer Choices:

1 1cos2β
2 1sin2β
3 (1cosβ)(1+cosβ)
4 (1sinβ)(1sinβ)
Answer: The correct expression is: .


ID is: 3605 Seed is: 9611

Proofs: trigonometric identities

Consider the following equation:

12cos2θ=(tanθ+1)(tanθ1)cos2θ

Prove that this equation is an identity. (Select the correct steps and reasons to complete the proof below.)

Answer:
Steps Reasons
(tanθ+1)(tanθ1)cos2θ Given
(tan2θ1)cos2θ
tan2θcos2θcos2θ
(sin2θcos2θ)cos2θcos2θ Quotient identity substitution
Cancel cos2θ factors
(1cos2θ)cos2θ Square identity substitution
Collect cos2θ terms

ID is: 3605 Seed is: 1799

Proofs: trigonometric identities

Consider the following equation:

3=3sinθcosθtanθ+3sin2θ

Prove that this equation is an identity. (Select the correct steps and reasons to complete the proof below.)

Answer:
Steps Reasons
3sinθcosθtanθ+3sin2θ Given
3sinθcosθ(sinθcosθ)+3sin2θ
3sinθcosθ(cosθsinθ)+3sin2θ Change division to multiplication
3cosθcosθ+3sin2θ Cancel sine factors
Combine cosine factors
Factorisation
3(sin2θ+cos2θ) Commutativity of addition
3(1) Square identity substitution
3


ID is: 3460 Seed is: 6102

Naming important identities

Here are three equations. All three are true maths equations.

Equation A tanθ=m
Equation B tanx=sinxcosx
Equation C sin2x+cos2x=1

Which of these equations is the quotient identity?

Answer: The quotient identity is .

ID is: 3460 Seed is: 464

Naming important identities

Here are three equations. All three are true maths equations.

Equation A tanx=sinxcosx
Equation B sin2x+cos2x=1
Equation C d=(x2x1)2+(y2y1)2

Which of these equations is the square identity?

Answer: The square identity is .


ID is: 3607 Seed is: 9213

Properties of a proof

  1. What does it mean to prove something in mathematics?

    Answer:

    To prove something in maths, one must write down with . The last step of the proof must as whatever you want to prove.

  2. The table below shows the proof for the identity:

    1cosθsinθ=sinθ1+cosθ

    However, the proof is not done yet: there are two steps and two reasons missing. Complete the proof by selecting the correct steps and reasons from the choices available.

    Answer:
    Steps Reasons
    Given
    Multiply by (1cosθ)(1cosθ)=1
    sinθ(1cosθ)(1+cosθ)(1cosθ)
    sinθ(1cosθ)(1cos2θ) Expand brackets
    sinθ(1cosθ)(sin2θ) Square identity substitution
    1cosθsinθ

ID is: 3607 Seed is: 347

Properties of a proof

  1. What does it mean to prove something in mathematics?

    Answer:

    To prove something in maths, one must write down with . The last step of the proof must as whatever you want to prove.

  2. The table below shows the proof for the identity:

    sinθ1cosθ=1+cosθsinθ

    However, the proof is not done yet: there are two steps and two reasons missing. Complete the proof by selecting the correct steps and reasons from the choices available.

    Answer:
    Steps Reasons
    sinθ1cosθ
    sinθ(1cosθ)(1+cosθ)(1+cosθ) Multiply by (1+cosθ)(1+cosθ)=1
    sinθ(1+cosθ)(1cosθ)(1+cosθ)
    sinθ(1+cosθ)(1cos2θ) Expand brackets
    Square identity substitution
    Cancel sinθ factors


ID is: 3585 Seed is: 6672

Proving an identity

  1. The definition of identity is an equation which is true for all values of the variable in the equation (as long as both sides are not undefined). Is the following equation an identity or not?

    x2+0x25=(x5)(x+5)
    Answer: The equation an identity.
  2. If you must prove an identity, what must you do?

    Answer:

    To prove something in maths, one must write down with . The last step of the proof must be whatever you want to prove.

  3. If you need to prove the following identity, which side of the equation should you start with?

    0=4+4tanx(cosxsinx)
    Answer: It is best to start with the of the equation.

ID is: 3585 Seed is: 1512

Proving an identity

  1. The definition of identity is an equation which is true for all values of the variable in the equation (as long as both sides are not undefined). Is the following equation an identity or not?

    2(x4)+2x=4x8
    Answer: The equation an identity.
  2. If you must prove an identity, what must you do?

    Answer:

    To prove something in maths, one must write down with . The last step of the proof must be whatever you want to prove.

  3. If you need to prove the following identity, which side of the equation should you start with?

    4+sin2x+cos2x=5
    Answer: It is best to start with the of the equation.


ID is: 3466 Seed is: 5260

Where does the quotient identity come from?

The quotient identity is:

tanθ=sinθcosθ

The two questions below are about the origins of this identity.

  1. Which of these choices is the origin of the quotient identity?

    Equation A sinθ=oppositehypotenuse
    Equation B tanθ=m
    Equation C tanθ=oppositeadjacent
    Equation D cos2θ=1sin2θ
    Answer:

    The origin of the quotient identity is .

  2. The table below shows a proof of the quotient identity. The proof consists of 4 steps and reasons. However, the proof has one missing step and one missing reason. Choose the correct step and reason to complete the proof.

    NOTE: In the proof, opp stands for opposite, adj stands for adjacent and hyp stands for hypotenuse.
    Answer:
    Steps Reasons
    tanθ=oppadj Definition of tangent ratio
    tanθ=oppadj1hyp1hyp
    tanθ=opphypadjhyp Fraction multiplication
    Definitions of sine and cosine ratios

ID is: 3466 Seed is: 8715

Where does the quotient identity come from?

The quotient identity is:

tanθ=sinθcosθ

The two questions below are about the origins of this identity.

  1. Which of these choices is the origin of the quotient identity?

    Equation A tanθ=oppositeadjacent
    Equation B a2+b2=c2
    Equation C tanθ=adjacentopposite
    Equation D cos2θ=1sin2θ
    Answer:

    The origin of the quotient identity is .

  2. The table below shows a proof of the quotient identity. The proof consists of 4 steps and reasons. However, the proof has one missing step and one missing reason. Choose the correct step and reason to complete the proof.

    NOTE: In the proof, opp stands for opposite, adj stands for adjacent and hyp stands for hypotenuse.
    Answer:
    Steps Reasons
    tanθ=oppadj Definition of tangent ratio
    Multiply by 1hyp1hyp=1
    tanθ=opphypadjhyp
    tanθ=sinθcosθ Definitions of sine and cosine ratios


ID is: 3459 Seed is: 732

Definition: identity

  1. In maths, an identity is something special. Complete the definition for identity below by choosing the correct missing words. Assume that the identity uses the variable x.

    Answer:

    An identity is which as long as both sides are defined.

  2. The following equation is not an identity:

    x2+3x10=0

    Why is this not an identity?

    Answer:

    The equation is not an identity because .


ID is: 3459 Seed is: 26

Definition: identity

  1. In maths, an identity is something special. Complete the definition for identity below by choosing the correct missing words. Assume that the identity uses the variable x.

    Answer:

    An identity is which as long as both sides are defined.

  2. The following equation is not an identity:

    y28y+15=0

    Why is this not an identity?

    Answer:

    The equation is not an identity because .



ID is: 3575 Seed is: 4150

Using the quotient identity to simplify an expression

Here is an expression which includes the tangent function.

6cosxtanx

Use the quotient identity to simplify this expression.

INSTRUCTION: If you need to type a trigonometric expression, put brackets around the angle. For example: sin(x) or cos(x).
Answer:

The simplified answer is: .

expression

ID is: 3575 Seed is: 4667

Using the quotient identity to simplify an expression

Here is an expression which includes the tangent function.

6cosxtanx

Use the quotient identity to simplify this expression.

INSTRUCTION: If you need to type a trigonometric expression, put brackets around the angle. For example: sin(x) or cos(x).
Answer:

The simplified answer is: .

expression


ID is: 4365 Seed is: 6814

Proving trig identities: difference of two squares

Adapted from DBE Nov 2016 Grade 11, P2, Q5.4
Maths formulas

Prove the following identity:

3tan2θsin2θ =3(sinθ1)(sinθ+1)

Kate has already completed the proof. But, two parts are missing.

LHS=3tan2θsin2θ=Block A=3cos2θ=Block B=3(sin2θ1)=3(sinθ1)(sinθ+1)=RHS

Select the most correct options for Block A and Block B to complete Kate's proof.

Answer:

Block A:

Block B:


ID is: 4365 Seed is: 5970

Proving trig identities: difference of two squares

Adapted from DBE Nov 2016 Grade 11, P2, Q5.4
Maths formulas

Prove the following identity:

1tan2θcos2θ =1(cosθ+1)(cosθ1)

Ikenna has already completed the proof. But, two parts are missing.

LHS=1tan2θcos2θ=Block A=1sin2θRHS=1(cosθ+1)(cosθ1)=1cos2θ1=Block B=1sin2θLHS=RHS

Select the most correct options for Block A and Block B to complete Ikenna's proof.

Answer:

Block A:

Block B:



ID is: 3651 Seed is: 6352

Examining identities

For what values of ψ is the following identity undefined within the domain ψ[360°;0°]?

8+3cos2ψ+2sin2ψy=11sin2ψy
Answer:

Within the domain ψ[360°;0°], the identity is undefined for ψ .


ID is: 3651 Seed is: 4629

Examining identities

For what values of ϕ is the following identity undefined within the domain ϕ[180°;180°]?

6cos2ϕ31+sinϕ=31sinϕ
Answer:

Within the domain ϕ[180°;180°], the identity is undefined for ϕ .



ID is: 1441 Seed is: 1445

Remembering the trigonometric identities

  1. Which of the expressions below belongs on the empty side of the equation?

    ??=tanα

    Answer Choices:

    A sinαcosα
    B sin2α+cos2α
    C 1
    Answer: The correct choice is: .
  2. What is the correct name for the identity in Question 1?

    Answer: The correct name is: .

ID is: 1441 Seed is: 6430

Remembering the trigonometric identities

  1. Which of the expressions below belongs on the empty side of the equation?

    ??=tanα

    Answer Choices:

    A sinαcosα
    B sin2α+cos2α
    C 1
    Answer: The correct choice is: .
  2. What is the correct name for the identity in Question 1?

    Answer: The correct name is: .

3. Reduction formula



ID is: 1509 Seed is: 5900

Simplifying trigonometric expressions with reduction formulas

Without using a calculator, determine the value of the following expression.

cos(420°)
INSTRUCTIONS:
  • Give your answer in surd form if necessary. You can type a surd like this: sqrt(10).
  • Do not use a calculator. While you can type the question into your calculator to get the answer, you will only get full marks in tests and exams if you show the necessary steps.
Answer: cos(420°)=
expression

ID is: 1509 Seed is: 8460

Simplifying trigonometric expressions with reduction formulas

Determine the value of the following without using a calculator.

sin(210°)
INSTRUCTIONS:
  • Give your answer in surd form if necessary. You can type a surd like this: sqrt(10).
  • Do not use a calculator. While you can type the question into your calculator to get the answer, you will only get full marks in tests and exams if you show the necessary steps.
Answer: sin(210°)=
expression


ID is: 1514 Seed is: 801

Applying reduction formulas

Simplify the following.

sin(90°α)tan(180°+α)cos(180°α)tan(180°α)
Answer: The simplified expression is: .
numeric

ID is: 1514 Seed is: 1644

Applying reduction formulas

Simplify the following.

2sin(90°α)sin(180°+α)cos(180°+α)sin(360°+α)
Answer: The simplified expression is: .
numeric


ID is: 1515 Seed is: 570

Applying reduction formulas

Simplify the following as much as possible, using only acute angles in your answer if necessary.

2tan(350°)cos2(370°)cos(190°)2cos(170°)
INSTRUCTION: If you need to type any trigonometric functions, put brackets around the angles, like this: sin(45).
Answer: The simplified expression is: .
one-of
type(expression.allow(['sin', 'cos']))

ID is: 1515 Seed is: 9165

Applying reduction formulas

Simplify the following as much as possible, using only acute angles in your answer if necessary.

sin2(205°)sin(155°)sin(745°)3sin(385°)
INSTRUCTION: If you need to type any trigonometric functions, put brackets around the angles, like this: sin(45).
Answer: The simplified expression is: .
one-of
type(expression.allow(['sin', 'cos']))


ID is: 1444 Seed is: 7652

Reduction formulas for the trigonometric functions

Reduce (simplify) the following trigonometric expression:

sin(360°+x)

For example, the expression cos(180°+x) can be reduced to: cos(x).

INSTRUCTION: If you need to type a trigonometric function, put brackets around the angle. For example: cos(x).
Answer: sin(360°+x)=
expression

ID is: 1444 Seed is: 8625

Reduction formulas for the trigonometric functions

Reduce (simplify) the following trigonometric expression:

cos(360°+x)

For example, the expression sin(180°+x) can be reduced to: sin(x).

INSTRUCTION: If you need to type a trigonometric function, put brackets around the angle. For example: sin(x).
Answer: cos(360°+x)=
expression


ID is: 1510 Seed is: 4050

Simplifying trigonometric expressions with reduction formulas

Given that tan(25°)=k, express the following in terms of k.

tan(385°)+4
INSTRUCTION: If you need to type a trigonometric function, put brackets around the angle, like this: sin(15).
Answer: tan(385°)+4=
expression

ID is: 1510 Seed is: 1351

Simplifying trigonometric expressions with reduction formulas

Simplify the following expression. The answer will include an acute angle.

sin(156°)
INSTRUCTION: If you need to type a trigonometric function, put brackets around the angle, like this: sin(15).
Answer: sin(156°)=
one-of
type(expression)


ID is: 4383 Seed is: 8442

Pythagorean problems

Adapted from DBE Nov 2016 Grade 11, P2, Q5.2
Maths formulas

Given: tan21°=m

Determine the following in terms of m, without using a calculator.

  1. sin21°

    Answer:
  2. sin111°

    Answer:
  3. tan2699°+tan2(21°)

    Answer: tan2699°+tan2(21°)=
    expression

ID is: 4383 Seed is: 2332

Pythagorean problems

Adapted from DBE Nov 2016 Grade 11, P2, Q5.2
Maths formulas

Given: tan31°=a

Determine the following in terms of a, without using a calculator.

  1. sin31°

    Answer:
  2. sin121°

    Answer:
  3. tan2689°+tan2(31°)

    Answer: tan2689°+tan2(31°)=
    expression


ID is: 3647 Seed is: 2827

Reduction formulas: "out of order" angles with negatives

Reduce (simplify) the following trigonometric expression:

sin(x360°)
INSTRUCTION: If you need to type a trigonometric function, put brackets around the angle. For example: cos(x).
Answer: sin(x360°)=
expression

ID is: 3647 Seed is: 4799

Reduction formulas: "out of order" angles with negatives

Reduce (simplify) the following trigonometric expression:

cos(x90°)
INSTRUCTION: If you need to type a trigonometric function, put brackets around the angle. For example: sin(x).
Answer: cos(x90°)=
expression


ID is: 1518 Seed is: 8365

Applying reduction formulas

Simplify the following as much as possible.

2sin(720°+A)cos(A)cos(A+180°)sin(360°+A)
INSTRUCTION: If you need to type any trigonometric functions, put brackets around the angles, like this: sin(45).
Answer: The simplified expression is: .
numeric

ID is: 1518 Seed is: 7093

Applying reduction formulas

Simplify the following as much as possible.

3cos(x)tan(x+360°)tan(720°x)cos(360°x)
INSTRUCTION: If you need to type any trigonometric functions, put brackets around the angles, like this: sin(45).
Answer: The simplified expression is: .
numeric


ID is: 4000 Seed is: 1935

Trigonometric expressions

Adapted from DBE Nov 2015 Grade 12, P2, Q5.2
Maths formulas

Given that sin(53°)=p.

Determine, in its simplest form, the value of each of the following in terms of p, without using a calculator:

  1. sin(307°)
  2. cos(53°)
  3. tan(127°)
INSTRUCTIONS:
  • If you need to type a trigonometric function, put brackets around the angle, like this: sin(15).
  • If you need to type a square root, it should look like this: sqrt(15).
Answer:
  1. sin(307°)=
  2. cos(53°)=
  3. The correct equation for tan(127°) is:
expression
expression

ID is: 4000 Seed is: 7455

Trigonometric expressions

Adapted from DBE Nov 2015 Grade 12, P2, Q5.2
Maths formulas

Given that cos(18°)=k.

Determine, in its simplest form, the value of each of the following in terms of k, without using a calculator:

  1. cos(162°)
  2. sin(18°)
  3. tan(162°)
INSTRUCTIONS:
  • If you need to type a trigonometric function, put brackets around the angle, like this: sin(15).
  • If you need to type a square root, it should look like this: sqrt(15).
Answer:
  1. cos(162°)=
  2. sin(18°)=
  3. The correct equation for tan(162°) is:
expression
expression


ID is: 4368 Seed is: 9717

Trig: Pythagorean problems in the Cartesian plane

Adapted from DBE Nov 2015 Grade 11, P2, Q5.1
Maths formulas

In the diagram below, D(30;y) is a point such that OD=34 and EO^D=α, where α is a reflex angle.

  1. Calculate the value of y.

    Answer:

    y=

    numeric
  2. Determine the value of each of the following without using a calculator.

    INSTRUCTION: Give your answer as a simplified fraction, e.g. 2/3 or -5/3.
    Answer:
    1. tanα=
    2. sin(360°+α)=
    3. cos(180°α)=
    numeric
    numeric
    numeric

ID is: 4368 Seed is: 510

Trig: Pythagorean problems in the Cartesian plane

Adapted from DBE Nov 2015 Grade 11, P2, Q5.1
Maths formulas

In the diagram below, P(21;y) is a point such that OP=29 and QO^P=β, where β is a reflex angle.

  1. Calculate the value of y.

    Answer:

    y=

    numeric
  2. Determine the value of each of the following without using a calculator.

    INSTRUCTION: Give your answer as a simplified fraction, e.g. 2/3 or -5/3.
    Answer:
    1. tanβ=
    2. cos(180°β)=
    3. sin(360°+β)=
    numeric
    numeric
    numeric


ID is: 1434 Seed is: 1645

Reduction formulas: using the CAST diagram

For the expression tan(180°x), which of the following is correct?

A tan(180°x)=+tan(x)
B tan(180°x)=tan(x)
TIP: Use what you know about the CAST diagram to decide if the right side of the equation should be positive or negative.
Answer: The correct equation is choice .

ID is: 1434 Seed is: 8174

Reduction formulas: using the CAST diagram

For the expression cos(180°y), which of the following is correct?

A cos(180°y)=+cos(y)
B cos(180°y)=cos(y)
TIP: Use what you know about the CAST diagram to decide if the right side of the equation should be positive or negative.
Answer: The correct equation is choice .


ID is: 3930 Seed is: 2011

Trigonometric expressions

Adapted from DBE Nov 2016 Grade 12, P2, Q5.1
Maths formulas

Answer the two questions below, using the fact that cos(16°)=p.

  1. Determine the following in terms of p without using a calculator:

    cos(196°)
    INSTRUCTIONS:
    • If you need to type a trigonometric function, put brackets around the angle, like this: sin(15).
    • If you need to type a square root, it should look like this: sqrt(15).
    Answer: cos(196°)=
    expression
  2. Determine the following in terms of p without using a calculator:

    sin(16°)
    INSTRUCTIONS:
    • If you need to type a trigonometric function, put brackets around the angle, like this: sin(15).
    • If you need to type a square root, it should look like this: sqrt(15).
    Answer: sin(16°)=
    expression

ID is: 3930 Seed is: 3327

Trigonometric expressions

Adapted from DBE Nov 2016 Grade 12, P2, Q5.1
Maths formulas

Answer the two questions below, using the fact that sin(10°)=t.

  1. Determine the following in terms of t without using a calculator:

    sin(190°)
    INSTRUCTIONS:
    • If you need to type a trigonometric function, put brackets around the angle, like this: sin(15).
    • If you need to type a square root, it should look like this: sqrt(15).
    Answer: sin(190°)=
    expression
  2. Determine the following in terms of t without using a calculator:

    cos(10°)
    INSTRUCTIONS:
    • If you need to type a trigonometric function, put brackets around the angle, like this: sin(15).
    • If you need to type a square root, it should look like this: sqrt(15).
    Answer: cos(10°)=
    expression


ID is: 1445 Seed is: 586

Reduction formulas: when the angle is 'out of order'

Reduce (simplify) the following trigonometric expression:

tan(x+180°)

For example, the expression sin(x+90°) can be reduced to: cos(x).

INSTRUCTION: If you need to type a trigonometric function, put brackets around the angle. For example: sin(x).
Answer: tan(x+180°)=
expression

ID is: 1445 Seed is: 5863

Reduction formulas: when the angle is 'out of order'

Reduce (simplify) the following trigonometric expression:

tan(x+360°)

For example, the expression sin(x+180°) can be reduced to: sin(x).

INSTRUCTION: If you need to type a trigonometric function, put brackets around the angle. For example: sin(x).
Answer: tan(x+360°)=
expression

4. Trigonometric equations



ID is: 4379 Seed is: 6428

Drawing and interpreting graphs of trig functions

Adapted from DBE Nov 2015 Grade 11, P2, Q6
Maths formulas

In the diagram below, the graph of g(x)=cos(x30°) is drawn for the interval 90°x225°.

  1. Redraw the graph of g. On the same system of axes, draw the graph of p(x)=sinx for the interval 90°x225°.

    Clearly show all intercepts with the axes, the coordinates of the turning points, and the end points of the graph.

    From your graph of p, read off and provide the coordinates asked for in the questions below.

    INSTRUCTION:

    Type brackets around the x- and y-values, and type a semicolon in between like this: (x; y)

    Answer:

    From the graph of p, give the coordinates of:

    1. the y-intercept of p:
    2. one x-intercept of p:
    3. one maximum turning point of p:
    4. one minimum turning point of p:
    coordinate
    one-of
    type(coordinate)
    one-of
    type(coordinate)
    one-of
    type(coordinate)
  2. Solve the equation cos(x30°)+sinx=0 on the interval 90°x225°.

    INSTRUCTIONS:
    • Round your answers to the nearest integer.
    • Do not use the degree symbol in your answers. It has been provided for you.
    Answer:

    x= °
    or
    x= °

    numeric
    numeric
  3. Determine the values of x in the interval 90°x225°, for which cos(x30°)+sinx0.

    INSTRUCTIONS:
    • Round your answers to the nearest integer.
    • Do not use the degree symbol in your answers. It has been provided for you.
    Answer:

    The correct interval which represents the values of x for which cos(x30°)+sinx0 is:

    Where:

    • a= °
    • b= °
    numeric
    numeric

ID is: 4379 Seed is: 983

Drawing and interpreting graphs of trig functions

Adapted from DBE Nov 2015 Grade 11, P2, Q6
Maths formulas

In the diagram below, the graph of g(x)=cos(x45°) is drawn for the interval 90°x225°.

  1. Redraw the graph of g. On the same system of axes, draw the graph of p(x)=sinx for the interval 90°x225°.

    Clearly show all intercepts with the axes, the coordinates of the turning points, and the end points of the graph.

    From your graph of p, read off and provide the coordinates asked for in the questions below.

    INSTRUCTION:

    Type brackets around the x- and y-values, and type a semicolon in between like this: (x; y)

    Answer:

    From the graph of p, give the coordinates of:

    1. the y-intercept of p:
    2. one x-intercept of p:
    3. one maximum turning point of p:
    4. one minimum turning point of p:
    coordinate
    one-of
    type(coordinate)
    one-of
    type(coordinate)
    one-of
    type(coordinate)
  2. Solve the equation cos(x45°)+sinx=0 on the interval 90°x225°.

    INSTRUCTIONS:
    • Round your answers to one decimal place.
    • Do not use the degree symbol in your answers. It has been provided for you.
    Answer:

    x= °
    or
    x= °

    numeric
    numeric
  3. Determine the values of x in the interval 90°x225°, for which cos(x45°)+sinx>0.

    INSTRUCTIONS:
    • Round your answers to one decimal place.
    • Do not use the degree symbol in your answers. It has been provided for you.
    Answer:

    The correct interval which represents the values of x for which cos(x45°)+sinx>0 is:

    Where:

    • a= °
    • b= °
    numeric
    numeric


ID is: 3974 Seed is: 2249

Graphs of trigonometric functions

Adapted from DBE Nov 2016 Grade 12, P2, Q6
Maths formulas

In the diagram, the graph of f(x)=sin2x is drawn for the interval x[180°;180°].

  1. Redraw the graph of f. On the same set of axes, draw the graph of g(x)=2cos2x for x[180°;180°].

    Clearly show all intercepts with the axes, the coordinates of the turning points, and the end points of the graph.

    From your graph of g, read off and provide the coordinates asked for in the questions below.

    INSTRUCTION: Give all coordinates in the form (x ; y). You must include the brackets, as well as the semi-colon.
    Answer:

    Give the coordinates of

    1. the y-intercept of g:
    2. any x-intercept of g:
    3. any turning point of g:
    coordinate
    one-of
    type(coordinate)
    one-of
    type(coordinate)
  2. What is the maximum value of f(x)1?

    Answer:

    The maximum value is:

    numeric
  3. Determine the general solution of f(x)=g(x).

    INSTRUCTION: Round your answer to two decimal places.
    Answer:

    The general solution is:
    x= ° +k°, where k is any .

    one-of
    type(numeric.abserror(0.01))
    numeric
  4. Hence, determine the values for x for which f(x)g(x) in the interval x[180°;0°].

    INSTRUCTIONS:
    • Round your answers to two decimal places.
    • Do not use the degree symbol in your answer. It has been provided for you.
    Answer:

    Choose the correct interval from the table below and determine the values of a and b.

    A a<x<b
    B axb
    C 180°x<aorb<x0°
    D 180°xaorbx0°

    The correct interval which represents the values of x for which f(x)g(x) is .

    Where:
    a= °
    b= °

    one-of
    type(numeric.abserror(0.01))
    one-of
    type(numeric.abserror(0.01))

ID is: 3974 Seed is: 4744

Graphs of trigonometric functions

Adapted from DBE Nov 2016 Grade 12, P2, Q6
Maths formulas

In the diagram, the graph of f(x)=3sin3x is drawn for the interval x[120°;120°].

  1. Redraw the graph of f. On the same set of axes, draw the graph of g(x)=2cos3x for x[120°;120°].

    Clearly show all intercepts with the axes, the coordinates of the turning points, and the end points of the graph.

    From your graph of g, read off and provide the coordinates asked for in the questions below.

    INSTRUCTION: Give all coordinates in the form (x ; y). You must include the brackets, as well as the semi-colon.
    Answer:

    Give the coordinates of

    1. the y-intercept of g:
    2. any x-intercept of g:
    3. any turning point of g:
    coordinate
    one-of
    type(coordinate)
    one-of
    type(coordinate)
  2. What is the maximum value of g(x)3?

    Answer:

    The maximum value is:

    numeric
  3. Determine the general solution of f(x)=g(x).

    INSTRUCTION: Round your answer to two decimal places.
    Answer:

    The general solution is:
    x= ° +k°, where k is any .

    one-of
    type(numeric.abserror(0.01))
    numeric
  4. Hence, determine the values for x for which f(x)>g(x) in the interval x[120°;0°].

    INSTRUCTIONS:
    • Round your answers to two decimal places.
    • Do not use the degree symbol in your answer. It has been provided for you.
    Answer:

    Choose the correct interval from the table below and determine the values of a and b.

    A a<x<b
    B axb
    C 120°x<aorb<x0°
    D 120°xaorbx0°

    The correct interval which represents the values of x for which f(x)>g(x) is .

    Where:
    a= °
    b= °

    one-of
    type(numeric.abserror(0.01))
    one-of
    type(numeric.abserror(0.01))


ID is: 1459 Seed is: 5550

Solving trigonometric equations: the general solution

Find the general solution for:

tanβ4=−7
INSTRUCTION: Round your answers to one decimal place, and if there is more than one answer, separate the answers with " ; " like this:
15,4+360k;164,6+360k
Answer: β= °kZ
list

ID is: 1459 Seed is: 5088

Solving trigonometric equations: the general solution

Find the general solution for:

sinθ+5=5,72
INSTRUCTION: Round your answers to one decimal place, and if there is more than one answer, separate the answers with " ; " like this:
15,4+360k;164,6+360k
Answer: θ= °kZ
list


ID is: 1497 Seed is: 8364

Solving quadratic trigonometric equations

Solve:

2cos2(θ)+5=7cos(θ)
INSTRUCTION: Find the general solution and write the answer to the nearest integer. If there are numerous answers, separate each solution by ";".
Answer: θ= °,kZ
list

ID is: 1497 Seed is: 4255

Solving quadratic trigonometric equations

Solve:

4sin2(x)+5=12sin(x)
INSTRUCTION: Give the general solution and write the answer to the nearest integer. If there are numerous answers, separate each solution by ";".
Answer: x= °,kZ
list


ID is: 1457 Seed is: 2663

The general solution for trigonometric equations

One of your classmates is solving this equation:

sinθ=−0,87

Her calculator tells her that the angle is θ=60°. For the general solution, what must she add onto this angle? (Take k as an element of the integers.)

ANSWER CHOICES:

A 360°k
B 0°k
C 180°k
D None of the above
Answer:

The other group of solutions is choice .


ID is: 1457 Seed is: 7952

The general solution for trigonometric equations

One of your classmates is solving this equation:

cosθ=0,5

Her calculator tells her that the angle is θ=60°. For the general solution, what must she add onto this angle? (Take k as an element of the integers.)

ANSWER CHOICES:

A 0°k
B 180°k
C 360°k
D None of the above
Answer:

The other group of solutions is choice .



ID is: 4361 Seed is: 6877

Trig equations: find the general solution by simplifying the expression first

Adapted from DBE Nov 2015 Grade 11, P2, Q5.3
Maths formulas

Consider the expression: 4cos(90°+B)tanB

  1. Simplify the expression to a single trigonometric term.

    Answer:
  2. Hence, determine the general solution of

    4cos(90°+2θ)tan2θ=1,28
    INSTRUCTION: Round your answer to two decimal places, if necessary.
    Answer:

    θ= ° +k ° ,kZ

    or

    θ= ° +k ° ,kZ

    one-of
    type(numeric.abserror(0.01))
    numeric
    one-of
    type(numeric.abserror(0.01))
    numeric

ID is: 4361 Seed is: 2281

Trig equations: find the general solution by simplifying the expression first

Adapted from DBE Nov 2015 Grade 11, P2, Q5.3
Maths formulas

Consider the expression: 1sin2P2sin(90°+P)

  1. Simplify the expression to a single trigonometric term.

    Answer:
  2. Hence, determine the general solution of

    1sin22y2sin(90°+2y)=0,29
    INSTRUCTION: Round your answer to two decimal places, if necessary.
    Answer:

    y= ° +k ° ,kZ

    or

    y= ° +k ° ,kZ

    one-of
    type(numeric.abserror(0.01))
    numeric
    one-of
    type(numeric.abserror(0.01))
    numeric


ID is: 1481 Seed is: 1026

Trigonometric equations: interval solutions

The general solution for the equation

cosx=12

is

x=120°+360°k;120°+360°k,kZ

Find all solutions to the equation in the interval x[0°;360°). (Hint: in this case there are less than three correct answers.)

INSTRUCTION: If there is more than one answer, separate the answers with " ; " like this:
45;30;30;45
If there are no answers in the interval, write No solution.
Answer: x= °
list

ID is: 1481 Seed is: 4130

Trigonometric equations: interval solutions

The general solution for the equation

sinβ=12

is

β=30°+360°k;150°+360°k,kZ

Find all solutions to the equation in the interval 0°β<90°. (Hint: in this case there are less than three correct answers.)

INSTRUCTION: If there is more than one answer, separate the answers with " ; " like this:
45;30;30;45
If there are no answers in the interval, write No solution.
Answer: β= °
list


ID is: 1458 Seed is: 8894

Solving trigonometric equations: the general solution

Find the general solution for:

2siny=2
INSTRUCTIONS:
  • Give your answers as integer values, with the variable k.
  • If there is more than one answer, separate the answers with ;.
    For example: 15+360k;165+360k.
  • If there is no answer, type no solution.
Answer: y= °kZ
list

ID is: 1458 Seed is: 3352

Solving trigonometric equations: the general solution

Solve the following equation. Give the general solution.

4cosθ=8
INSTRUCTIONS:
  • Give your answers as integer values, with the variable k.
  • If there is more than one answer, separate the answers with ;.
    For example: 15+360k;165+360k.
  • If there is no answer, type no solution.
Answer: θ= °kZ
string


ID is: 1483 Seed is: 5459

Trigonometric equations: finding intervals from general solutions

  1. Given the equation x=30°+180°k,kZ for the values of angle x, determine the value of the angle if k=1.

    Answer:

    The value of the angle x if k=1 is °

    numeric
  2. Is this angle in the interval x[90°;0°), or not?

    Answer:

    The angle x is in the interval x[90°;0°)?


ID is: 1483 Seed is: 5297

Trigonometric equations: finding intervals from general solutions

  1. Given the equation x=135°+360°k,kZ for the values of angle x, determine the value of the angle if k=0.

    Answer:

    The value of the angle x if k=0 is °

    numeric
  2. Is this angle in the interval x(270°;0°], or not?

    Answer:

    The angle x is in the interval x(270°;0°]?



ID is: 1470 Seed is: 684

Facts about trigonometric equations

You run across a friend who is looking at this equation:

cosθ=0,71
They are not sure what to expect, so you want to give them a bit of help. Which of the following hints can you give them?

Answer:

The hint I can give them is: .


ID is: 1470 Seed is: 3157

Facts about trigonometric equations

You run across a friend who is looking at this equation:

cosθ=1
They are not sure what to expect, so you want to give them a bit of help. Which of the following hints can you give them?

Answer:

The hint I can give them is: .



ID is: 1498 Seed is: 3570

Interval solutions with complex angles

Find the solution for the equation below if 90°<x<0°:

2tan(2x15°)+1,5=6,7
INSTRUCTION: Round your answers to two decimal places, and if there is more than one answer, separate the answers with the " ; " symbol.
Answer: x= °
list

ID is: 1498 Seed is: 4382

Interval solutions with complex angles

Find the solution for the equation below if θ(180°;180°):

3cos(3θ+45°)+2,5=4,18
INSTRUCTION: Round your answers to two decimal places, and if there is more than one answer, separate the answers with the " ; " symbol.
Answer: θ= °
list


ID is: 4385 Seed is: 3921

General solutions to trigonometric equations

Adapted from DBE Nov 2016 Grade 11, P2, Q5.5
Maths formulas

Determine the general solution for the following:

3sinx=2sinxcosx
Answer:

sinx=

x= ° +k180°

OR

cosx=

x= ° +k360°
or
x= ° +k360°

where k is any .

one-of
type(numeric.abserror(1.0))
one-of
type(numeric.abserror(1.0))
one-of
type(numeric.abserror(1.0))

ID is: 4385 Seed is: 6548

General solutions to trigonometric equations

Adapted from DBE Nov 2016 Grade 11, P2, Q5.5
Maths formulas

Determine the general solution for the following:

3cosx=2sinxcosx
Answer:

cosx=

x= ° +k180°

OR

sinx=

x= ° +k360°
or
x= ° +k360°

where k is any .

one-of
type(numeric.abserror(1.0))
one-of
type(numeric.abserror(1.0))
one-of
type(numeric.abserror(1.0))


ID is: 1471 Seed is: 2623

General solutions with complex angles

Find the general solution for:

3cos(2y60°)+3=4,32
INSTRUCTION: Round your answers to two decimal places, and if there is more than one answer, separate the answers with the " ; " symbol.
Answer: y= °,kZ
list

ID is: 1471 Seed is: 7410

General solutions with complex angles

Solve the following equation. Give the general solution.

3sin(2θ90°)+5=6,8
INSTRUCTION: Round your answers to two decimal places, and if there is more than one answer, separate the answers with the " ; " symbol.
Answer: θ= °,kZ
list


ID is: 1482 Seed is: 9275

Trigonometric equations: interval solutions

Solve the equation sinβ=32 in the interval 180°β<270°.

INSTRUCTION: If there are multiple answers, separate the answers with " ; ". For example:
330;210;30
If there are no answers in the interval, write No solution.
Answer: β= °
list

ID is: 1482 Seed is: 8264

Trigonometric equations: interval solutions

Solve the equation cosα=12 in the interval α(360°;360°].

INSTRUCTION: If there are multiple answers, separate the answers with " ; ". For example:
330;210;30
If there are no answers in the interval, write No solution.
Answer: α= °
list

5. Area, sine, and cosine rules

5. The area rule



ID is: 3564 Seed is: 9302

Area in terms of the sine ratio

  1. What does is mean to find an answer "in terms of x"?

    Answer: It means the answer .
  2. Determine the area A of the triangle shown in terms of sinC.

    Answer: A=
    expression

ID is: 3564 Seed is: 7271

Area in terms of the sine ratio

  1. What does is mean to find an answer "in terms of x"?

    Answer: It means the answer .
  2. Determine the area A of the triangle shown in terms of sinC.

    Answer: A=
    expression


ID is: 3550 Seed is: 9305

The area rule: finding mistakes

One of your classmates is working on a trigonometry worksheet about the area rule. The worksheet has a triangle with sides 4, 6, and 7, and one angle labelled 59°. He wrote the area rule equation below the triangle diagram. But they made a mistake and the equation is incorrect.

Incorrect equation:

A=12(6)(7)sin59°

What should your classmate do to correct the equation?

Answer:

To make the equation correct, he should .


ID is: 3550 Seed is: 5877

The area rule: finding mistakes

One of your classmates is working on a trigonometry worksheet about the area rule. The worksheet has a triangle with sides 7, 8, and 9, and one angle labelled 60°. She wrote the area rule equation below the triangle diagram. But they made a mistake and the equation is incorrect.

Incorrect equation:

A=12(9)(8)sin60°

What should your classmate do to correct the equation?

Answer:

To make the equation correct, she should .



ID is: 3555 Seed is: 6081

Area formulas: old and new

The area rule for a triangle is:

A=12absinθ

You should also recognise this formula for the area of a triangle:

A=12bh

The two statements below are about the relationship between these equations. Complete the statements by selecting the best choice for each statement.

Answer:
  • For the area rule, the angle between a and b θ, but the angle between b and h must be perpendicular.
  • If we substitute θ=90° into the area rule it becomes the same as .

ID is: 3555 Seed is: 1414

Area formulas: old and new

The area rule for a triangle is:

A=12absinθ

You should also recognise this formula for the area of a triangle:

A=12bh

The two statements below are about the relationship between these equations. Complete the statements by selecting the best choice for each statement.

Answer:
  • For the area rule, the angle between a and b θ, but the angle between b and h must be perpendicular.
  • If we substitute θ=90° into the area rule it becomes the same as .


ID is: 3527 Seed is: 8964

Working with the area rule

Consider a triangle MNP with NP¯=6,92 cm and PM¯=7,48 cm. The area of the triangle is 21,95 cm2. This is sketched below, but the diagram is not drawn to scale.

There are two possible values for P^. Determine both values for P^.

INSTRUCTION: Round your answers to the hundredths place (two decimal places). You can type your answers in any order.
Answer: P^= ° or °
numeric
numeric

ID is: 3527 Seed is: 8208

Working with the area rule

Consider a triangle MNP with NP¯=7,66 mm and PM¯=10 mm. The area of the triangle is 31,75 mm2. This is sketched below, but the diagram is not drawn to scale.

There are two possible values for P^. Find both values for P^.

INSTRUCTION: Round your answers to the hundredths place (two decimal places). You can type your answers in any order.
Answer: P^= ° or °
numeric
numeric


ID is: 3557 Seed is: 2324

The flexibility of the area rule

The diagram below shows a triangle with the angles x^, y^, and z^. The lengths of the sides are 9, 7, and 6 units. Answer the two questions below about this triangle.

  1. For this triangle, we can enter the area rule in three different ways. Here are two of these:

    A=12(9)(7)siny^A=12(7)(6)sinz^

    Which of the choices below shows the third correct way we can write the area rule for this triangle?

    A A=12(6)(9)sinz^
    B A=12(6)(9)sinx^
    C A=12(7)(6)sinx^
    D A=12(6)(7)siny^
    Answer: The third option for the area rule is choice .
  2. Suppose we find out that z^=89° and that y^=41°. Use this information and the equations from Question 1 to calculate the area of the triangle.

    INSTRUCTION: Round your answer to 2 decimal places.
    Answer: The area is square units.
    one-of
    type(numeric.abserror(0.01))

ID is: 3557 Seed is: 4354

The flexibility of the area rule

The diagram below shows a triangle with the angles y^, x^, and z^. The lengths of the sides are 10, 12, and 8 units. Answer the two questions below about this triangle.

  1. For this triangle, we can enter the area rule in three different ways. Here are two of these:

    A=12(8)(10)siny^A=12(10)(12)sinx^

    Which of the choices below shows the third correct way we can write the area rule for this triangle?

    A A=12(8)(10)sinz^
    B A=12(8)(10)sinx^
    C A=12(12)(8)sinz^
    D A=12(12)(8)siny^
    Answer: The third option for the area rule is choice .
  2. Suppose we find out that y^=82° and that x^=41°. Use this information and the equations from Question 1 to calculate the area of the triangle.

    INSTRUCTION: Round your answer to 2 decimal places.
    Answer: The area is square units.
    one-of
    type(numeric.abserror(0.01))


ID is: 3529 Seed is: 795

Find the sine ratio

The figure below shows parallelogram WXYZ. Side YZ¯ is 5 cm long, and side WZ¯ is 8 cm long, as labelled. The angle at vertex W is labelled ψ, and the area of the parallelogram is 29 cm2. The diagram is not drawn to scale.

Determine the value of sinψ.

INSTRUCTION: Give your answer as a fraction.
Answer: The value of sinψ is .
numeric

ID is: 3529 Seed is: 8257

Find the sine ratio

The figure below shows parallelogram KLMN. Side MN¯ is 19 m long, and side KN¯ is 8 m long, as labelled. The angle at vertex M is labelled γ, and the area of the parallelogram is 32 m2. The diagram is not drawn to scale.

Determine the value of sinγ.

INSTRUCTION: Give your answer as a fraction.
Answer: The value of sinγ is .
numeric


ID is: 3562 Seed is: 1859

Area rule to the rescue

Consider a triangle ZXY, shown below. Two angles are given: Z=65,7° and Y=57,7°. The sides of the triangle are ZX¯=5,6 cm, YZ¯=2d+6, and XY¯=6d+2, where dR. The triangle may or may not be drawn to scale.

Determine the length of side XY¯. Then select the correct units for the length of the side.

INSTRUCTION:
  • Round your answer to 2 decimal places.
  • If there is no answer, type No solution in the answer box and select No units for the units.
Answer: XY¯=
one-of
type(numeric.abserror(0.004))

ID is: 3562 Seed is: 7402

Area rule to the rescue

Consider a triangle XZY, shown below. Two angles are given: X=69,1° and Y=66,4°. The sides of the triangle are XZ¯=6,3 mm, YX¯=5d+9, and ZY¯=6d+7, where dR. The triangle may or may not be drawn to scale.

Determine the length of side ZY¯. Then select the correct units for the length of the side.

INSTRUCTION:
  • Round your answer to 2 decimal places.
  • If there is no answer, type No solution in the answer box and select No units for the units.
Answer: ZY¯=
one-of
type(numeric.abserror(0.004))


ID is: 3501 Seed is: 4610

Using the area rule

The figure below is drawn to scale, and it shows a triangle. One angle is labelled 83°. And the sides next to that angle are 10 and 6.

Determine the area of the triangle.

INSTRUCTION: Round your answer to two decimal places if necessary.
Answer:

The area of the triangle is square units.

numeric

ID is: 3501 Seed is: 544

Using the area rule

The figure below is drawn to scale, and it shows a triangle. One angle is labelled 54°. And the sides next to that angle are 11 and 8.

Determine the area of the triangle.

INSTRUCTION: Round your answer to two decimal places if necessary.
Answer:

The area of the triangle is square units.

numeric


ID is: 3502 Seed is: 5177

Area rule calculations

An isosceles triangle, ΔABC, has equal sides AB¯ and AC¯. They are both 19 mm long. The third side is 18 mm long. The angle at C is 62°. The figure below shows this triangle. The figure is drawn to scale.

What is the triangle's area?

INSTRUCTION:
  • Round your answer to two decimal places if necessary.
  • Do not type units with your answer (type only the number).
Answer: The area of the triangle is mm2.
numeric

ID is: 3502 Seed is: 9631

Area rule calculations

The figure below shows an isosceles triangle, ΔDEF. Side EF¯ has length 18 cm while the other two sides are both 19 cm. One angle is known: F^=62°. The figure is drawn to scale.

Find the area of triangle DEF.

INSTRUCTION:
  • Round your answer to two decimal places if necessary.
  • Do not type units with your answer (type only the number).
Answer: The area of the triangle is cm2.
numeric


ID is: 3565 Seed is: 5769

Area in terms of a variable

The triangle below is drawn to scale. It has sides with lengths 5, 10, and an unknown length, 9f3. Two angles are given, A^=24° and C^=23°.

  1. Which of the following is a correct expression for the area of this triangle in terms of f with all numeric values rounded to two decimal places?

    A A=17,58f5,86
    B A=13,73f9,57
    C A=21,83f2,59
    Answer: The correct expression is choice .
  2. Determine the value of f.

    INSTRUCTION: Use the rounded answer from Question 1. Round your answer to two decimal places.
    Answer: f=
    one-of
    type(numeric.abserror(0.004))

ID is: 3565 Seed is: 3194

Area in terms of a variable

The triangle below is drawn to scale. It has sides with lengths 7, 10, and an unknown length, 8a2. Two angles are given, B^=32° and C^=46°.

  1. Which of the following is a correct expression for the area of this triangle in terms of a with all numeric values rounded to two decimal places?

    A A=32,77a4,07
    B A=28,77a7,19
    C A=25,27a10,35
    Answer: The correct expression is choice .
  2. Determine the value of a.

    INSTRUCTION: Use the rounded answer from Question 1. Round your answer to two decimal places.
    Answer: a=
    one-of
    type(numeric.abserror(0.004))


ID is: 1557 Seed is: 7660

Using the area rule

The figure below shows a triangle, which is drawn to scale. The sides are lengths 10,4 mm, 12,4 mm, and 12,4 mm, and the angle 65° is labelled.

  1. Determine the area of the triangle.
  2. Identify the correct units for the answer from the choices in the list below.
INSTRUCTION: Round your answer to one decimal place if necessary.
Answer:

The area of the triangle is .

numeric

ID is: 1557 Seed is: 1023

Using the area rule

The figure below shows a triangle, which is drawn to scale. The sides are lengths 9,3 cm, 9,3 cm, and 13,1 cm, and the angle 45° is labelled.

  1. Determine the area of the triangle.
  2. Identify the correct units for the answer from the choices in the list below.
INSTRUCTION: Round your answer to one decimal place if necessary.
Answer:

The area of the triangle is .

numeric


ID is: 3531 Seed is: 8846

Finding area pieces

Triangle ΔABC is inscribed in a circle. This means the vertices of the triangle lie on the circle's circumference. The radius of the circle is 4,2 m. The sides of the triangle are AC¯=6 m, CB¯=8,3 m, and BA¯=5,1 m. B^=46° and C^=38°.

Determine the area of the shaded region, the area inside the circle but outside of the triangle. Then select the correct units for the answer.

INSTRUCTION: Round your answer to three decimal places.
Answer: The shaded area is .
one-of
type(numeric.abserror(0.0004))

ID is: 3531 Seed is: 2048

Finding area pieces

Triangle ΔABC is inscribed in a circle. This means the vertices of the triangle lie on the circle's circumference. The radius of the circle is 3,7 m. The sides of the triangle are AB¯=6 m, BC¯=7,3 m, and CA¯=5,0 m. C^=55° and B^=43°.

Determine the area of the shaded region, the area inside the circle but outside of the triangle. Then select the correct units for the answer.

INSTRUCTION: Round your answer to three decimal places.
Answer: The shaded area is .
one-of
type(numeric.abserror(0.0004))


ID is: 3533 Seed is: 4531

Working with triangles

Two triangles share a common side, as shown below. The triangles have a common side, JK¯= 5,95 units. The sides around the perimeter have the lengths JG¯= 5,23 units, GK¯= 4,44 units, KH¯= 5,85 units, and HJ¯= 4,96 units. There are two angles given: HKJ=49,7° and JGK=75,3°. Note that the diagram is not drawn to scale.

  1. Determine the area of ΔJGK.

    INSTRUCTION: Round your answer to two decimal places.
    Answer: The area of ΔJGK is square units.
    numeric
  2. If it now given that HJK=64,1°, determine the angle JHK.

    Answer: JHK= °
    numeric

ID is: 3533 Seed is: 7648

Working with triangles

Two triangles share a common side, as shown below. The triangles have a common side, KJ¯= 6,88 units. The sides around the perimeter have the lengths KH¯= 4,74 units, HJ¯= 6,43 units, JG¯= 4,8 units, and GK¯= 4,55 units. There are two angles given: KGJ=94,5° and KJH=41,5°. Note that the diagram is not drawn to scale.

  1. Determine the area of ΔKGJ.

    INSTRUCTION: Round your answer to two decimal places.
    Answer: The area of ΔKGJ is square units.
    numeric
  2. If it now given that JKH=64°, determine the angle KHJ.

    Answer: KHJ= °
    numeric


ID is: 3561 Seed is: 7786

Find the triangle's area

Suppose that a triangle XYZ includes angle X^=134°. The side of the triangle across from this angle is YZ¯=3,08 cm long. The sides adjacent to this angle are ZX¯=5,29 cm long and XY¯=25g236. As a function of g, the area of the triangle is:

A(g)=5g+6
  1. Calculate the numerical value of the triangle's area.

    INSTRUCTION: Round your answer to two decimal places.
    Answer: The area is cm2.
    one-of
    type(numeric.abserror(0.004))
  2. In Question 1, the area function for the triangle is linear, which means the area is related to g as shown in this graph:

    This shows that as g increases, the area increases. Why does this happen?

    Answer: When g increases, .

ID is: 3561 Seed is: 3600

Find the triangle's area

Suppose that a triangle ZXY includes angle Z^=53°. The side of the triangle across from this angle is XY¯=2,55 mm long. The sides adjacent to this angle are YZ¯=2,38 mm long and ZX¯=6g+5. As a function of g, the area of the triangle is:

A(g)=42g2+35g
  1. Calculate the numerical value of the triangle's area.

    INSTRUCTION: Round your answer to two decimal places.
    Answer: The area is mm2.
    one-of
    type(numeric.abserror(0.004))
  2. In Question 1, the area function for the triangle is quadratic, which means the area is related to g as shown in this graph:

    This shows that as g increases, the area increases. Why does this happen?

    Answer: When g increases, .


ID is: 1435 Seed is: 3815

The area rule: identifying the correct sides and angles

The triangle below has sides labelled x,y, and z, and angles labelled X,Y, and Z.

For this triangle, the area rule says:

AreaΔ=12(?)zsinY

Which of the following choices belongs in the place of the '?' in the formula?

Answer: The missing value is: .

ID is: 1435 Seed is: 8363

The area rule: identifying the correct sides and angles

The triangle below has sides labelled a,b, and c, and angles labelled A,B, and C.

For this triangle, the area rule says:

AreaΔ=12acsin(?)

Which of the following choices belongs in the place of the '?' in the formula?

Answer: The missing value is: .


ID is: 3563 Seed is: 6478

We're bringing perimeter back!

Triangle YZX has a perimeter of 127 cm. The sides of the triangle are YZ¯=9 cm, ZX¯=y+8 and XY¯=8y8, where yQ. Two angles are given: Y=55,2° and X=20,5°. Determine the area of the triangle. Then select the correct units for the answer.

INSTRUCTION: Round your answer to two decimal places.
Answer: Area =
one-of
type(numeric.abserror(0.004))

ID is: 3563 Seed is: 7021

We're bringing perimeter back!

Triangle YXZ has a perimeter of 148 mm. The sides of the triangle are YX¯=7 mm, XZ¯=x6 and ZY¯=8x3, where xQ. Two angles are given: Y=76,5° and Z=39,6°. Determine the area of the triangle. Then select the correct units for the answer.

INSTRUCTION: Round your answer to two decimal places.
Answer: Area =
one-of
type(numeric.abserror(0.004))

6. The sine rule



ID is: 1556 Seed is: 763

Using the sine rule

Calculate the length of the unknown side in the triangle below. One side of the triangle has a length of 11,5. Two angles are labelled, with values of 73° and 48°. The figure below is drawn to scale.

TIP: You can compare the answer you get to the diagram because it is drawn to scale. The side you need is shorter than the given side 11,5 so check your answer before you submit it to make sure it agrees with the figure.
INSTRUCTION: Round your answer to the tenths place (one decimal) if necessary.
Answer: The length of the missing side is units.
numeric

ID is: 1556 Seed is: 9684

Using the sine rule

Given a triangle with an angle of 62° and two sides with lengths 7,4 and 11,1, find the missing angle, indicated by the question mark in the figure. The figure below is drawn to scale.

TIP: You can compare the answer you get to the diagram because it is drawn to scale. The angle that you want is acute, so check your answer before you submit it to make sure it agrees with the figure.
INSTRUCTION: Round your answer to the tenths place (one decimal) if necessary.
Answer: The missing angle is °
numeric


ID is: 3569 Seed is: 6118

Finding an angle with the sine rule

In ΔPRQ, Q^=104°, QP=4,8 cm, and PR=6,9 cm.

Calculate the size of R^.

INSTRUCTION: Round the answer to two decimal places.
Answer: R^= °
numeric

ID is: 3569 Seed is: 8794

Finding an angle with the sine rule

In ΔWXY, Y^=95°, YW=5,8 cm, and WX=9 cm.

Calculate the size of X^.

INSTRUCTION: Round the answer to two decimal places.
Answer: X^= °
numeric


ID is: 3570 Seed is: 6834

Sine rule problems

In the figure below, WZ^X=m, ZX^W=n, and XZ=d. XZ and WY are perpendicular to YX.

Determine the distance WX in terms of m, n, and d.

Answer: WX=
expression

ID is: 3570 Seed is: 4803

Sine rule problems

In the figure below, QS^R=a, SR^Q=b, and RS=h. RS and QP are perpendicular to PR.

Determine the distance SQ in terms of a, b, and h.

Answer: SQ=
expression


ID is: 3558 Seed is: 2431

Algebra meets the sine rule

In triangle EFD below, two angles are given: E^=71° and D^=68°. Side DE¯ and FD¯ have lengths 9y+7 cm and 8y+2 cm respectively, where yR. The third side, EF¯, is 20 cm long. The figure may or may not be drawn to scale.

  1. Determine the value of y.

    INSTRUCTION: Round your answer to two decimal places.
    Answer: y=
    one-of
    type(numeric.abserror(0.004))
  2. Hence determine the length of DE¯.

    INSTRUCTION:
    • Use the rounded answer from Question 1.
    • Round your answer to two decimal places if necessary.
    Answer: DE¯= cm
    one-of
    type(numeric.noerror)

ID is: 3558 Seed is: 8994

Algebra meets the sine rule

In triangle ACB below, two angles are given: A^=60° and B^=87°. Side BA¯ and CB¯ have lengths 6x2 cm and 4x+2 cm respectively, where xR. The third side, AC¯, is 17 cm long. The figure may or may not be drawn to scale.

  1. What is the value of x?

    INSTRUCTION: Round your answer to two decimal places.
    Answer: x=
    one-of
    type(numeric.abserror(0.004))
  2. Hence determine the length of BA¯.

    INSTRUCTION:
    • Use the rounded answer from Question 1.
    • Round your answer to two decimal places if necessary.
    Answer: BA¯= cm
    one-of
    type(numeric.noerror)


ID is: 3571 Seed is: 7771

The flexibility of the sine rule

The sine rule describes the relationship between the sides and angles in any triangle. The ratio of any side to the sine of the angle opposite from that side is the same for all three side-angle pairs in the triangle. For example, consider this triangle, which has sides z, x, and y, and angles F, H, and G. Two of the three equal ratios are shown below the triangle.

ysinH=zsinG=third ratio

Which of the choices below is the third equal ratio for this triangle?

A ysinF
B xsinF
C xsinH
Answer: The third ratio is choice .

ID is: 3571 Seed is: 148

The flexibility of the sine rule

The sine rule describes the relationship between the sides and angles in any triangle. The ratio of any side to the sine of the angle opposite from that side is the same for all three side-angle pairs in the triangle. For example, consider this triangle, which has sides p, m, and n, and angles H, F, and G. Two of the three equal ratios are shown below the triangle.

nsinF=psinG=third ratio

Which of the choices below is the third equal ratio for this triangle?

A psinH
B msinG
C msinH
Answer: The third ratio is choice .


ID is: 3580 Seed is: 4089

The sine rule: the ambiguous case

  1. In ΔLKJ, LJ¯=60 and JK¯=74. The angle across from side LJ¯ is K^=52,8°. Calculate the size of the angle across from JK¯, L^, if it is obtuse (greater than 90°).

    INSTRUCTION: Round your answers to two decimal places.
    Answer:

    The angle is °.

    numeric
  2. When you use the sine rule, sometimes you get the ambiguous case, which leads to two solutions. But Question 1 was not ambiguous: it has only one solution. Why is this the case?

    Answer:

    Question 1 was not ambiguous because .


ID is: 3580 Seed is: 1853

The sine rule: the ambiguous case

  1. In ΔMNP, NM¯=49 and PN¯=80. The angle across from side NM¯ is P^=37°. Calculate the size of the angle across from PN¯, M^, if it is obtuse (greater than 90°).

    INSTRUCTION: Round your answers to two decimal places.
    Answer:

    The angle is °.

    numeric
  2. When you use the sine rule, sometimes you get the ambiguous case, which leads to two solutions. But Question 1 was not ambiguous: it has only one solution. Why is this the case?

    Answer:

    Question 1 was not ambiguous because .



ID is: 3504 Seed is: 1844

Finding a length with the sine rule

The diagram below shows a triangle with two angles and one side labelled. The given angles are 36° and 44°. The side across from the 36° angle is 6. Determine the length of the side across from the 44° angle. The figure is drawn to scale.

TIP: This diagram is drawn to scale so you can compare the answer you get to the diagram. The side you need is longer than the given side 6. Check your answer before you submit it to make sure it agrees with the figure.
INSTRUCTION: Round your answer to two decimal places if necessary.
Answer: The length of the side is .
numeric

ID is: 3504 Seed is: 9479

Finding a length with the sine rule

The diagram below shows a triangle with two angles and one side labelled. The given angles are 92° and 24°. The side across from the 92° angle is 10. Determine the length of the side across from the 24° angle. The figure is drawn to scale.

TIP: This diagram is drawn to scale so you can compare the answer you get to the diagram. The side you need is shorter than the given side 10. Check your answer before you submit it to make sure it agrees with the figure.
INSTRUCTION: Round your answer to two decimal places if necessary.
Answer: The length of the side is .
numeric


ID is: 3551 Seed is: 2688

The sine rule: finding mistakes

In maths class you are learning about the sine rule. You are working on a problem in which the triangle has sides 4, 5, and 7, and two angles labelled A and B. You write down the sine rule equation for this triangle, as shown below. But your teacher walks by and tells you that there is a mistake in the equation.

Incorrect equation:

sinB4=sinA7

What should you do to correct the equation?

Answer:

To correct the equation you should .


ID is: 3551 Seed is: 2685

The sine rule: finding mistakes

In maths class you are learning about the sine rule. You are working on a problem in which the triangle has sides 7, 8, and 10, and two angles labelled A and B. You write down the sine rule equation for this triangle, as shown below. But your teacher walks by and tells you that there is a mistake in the equation.

Incorrect equation:

10sinB=8sinA

What should you do to correct the equation?

Answer:

To correct the equation you should .



ID is: 3559 Seed is: 8507

Find the missing length

In triangle BAC, two angles are known: B^=79° and A^=57°. Sides AC¯ and CB¯ have lengths 10d2+4d and 5d+2 respectively, where dR. All lengths are non-zero and have units of cm.

Find then length of CB¯.

INSTRUCTION: Round your answer to two decimal places.
Answer: The length of CB¯ is cm.
numeric

ID is: 3559 Seed is: 8894

Find the missing length

In triangle FDE, two angles are known: F^=66° and D^=45°. Sides DE¯ and EF¯ have lengths 15d2+9d and 5d+3 respectively, where dR. All lengths are non-zero and have units of cm.

Compute then length of EF¯.

INSTRUCTION: Round your answer to two decimal places.
Answer: The length of EF¯ is cm.
numeric


ID is: 3579 Seed is: 8780

The sine rule: the ambiguous case

The diagram below shows ΔKJL. The diagram is not to scale. Side LJ=9,5 cm and side JK=13,4 cm. One angle is given: K^=43,6°.

  1. Determine the values for L^. There are two correct answers.

    INSTRUCTION:
    • You can type your answers in any order.
    • Round your answers to two decimal places.
    Answer:

    The possible answers are ° and °.

    numeric
    numeric
  2. If you are now told that L^ is obtuse, which of the answers to Question 1 is the correct answer?

    Answer:

    The correct answer must be °.

    numeric

ID is: 3579 Seed is: 6068

The sine rule: the ambiguous case

The diagram below shows ΔPMN. The diagram is not to scale. Side MP=9,5 cm and side PN=13,4 cm. One angle is given: N^=39,2°.

  1. Determine the values for M^. There are two correct answers.

    INSTRUCTION:
    • You can type your answers in any order.
    • Round your answers to two decimal places.
    Answer:

    The possible answers are ° and °.

    numeric
    numeric
  2. If you are now told that M^ is acute, which of the answers to Question 1 is the correct answer?

    Answer:

    The correct answer must be °.

    numeric


ID is: 1433 Seed is: 5382

The sine rule: identifying the correct sides and angles

The triangle below has sides labelled x,y, and z, and angles labelled X,Y, and Z.

For this triangle, the sine rule states:

zsin?=ysinY

Which of the following choices belongs in the place of the '?' in the formula?

Answer: The missing symbol is: .

ID is: 1433 Seed is: 1943

The sine rule: identifying the correct sides and angles

The triangle below has sides labelled a,b, and m, and angles labelled α,β, and θ.

For this triangle, the sine rule states:

?sinα=msinθ

Which of the following choices belongs in the place of the '?' in the formula?

Answer: The missing symbol is: .


ID is: 3581 Seed is: 9755

Connecting the sine rule to right triangles

  1. The sine rule applies to any triangle. This includes right-angled triangles. Which statement below identifies what happens to the sine rule if one of the angles is 90°?

    Choices If an angle is 90°, the sine rule...
    A will not work anymore.
    B becomes the same as the theorom of Pythagoras.
    C becomes the sine ratio (opposite & hypotenuse).
    D only uses the reciprocals.
    Answer:

    The correct statement is choice .

  2. The table below shows (proves) that if we use the sine rule with a 90° angle it simplifies into the sine ratio for right-angled triangles. However, there is one missing step and one missing reason. Complete the proof by choosing the correct options from the lists available.

    Answer:
    Steps Reasons
    Given
    Sine rule for ΔXYZ
    sinXx=sin(90°)z
    sinXx=1z Evaluate sin90°=1
    sinX=xz Multiply by x
    sinX=oppositehypotenuse Definition of opposite & hypotenuse

ID is: 3581 Seed is: 1248

Connecting the sine rule to right triangles

  1. The sine rule applies to any triangle. This includes right-angled triangles. Which statement below identifies what happens to the sine rule if one of the angles is 90°?

    Choices If an angle is 90°, the sine rule...
    A will give two answers due to the ambiguous case.
    B becomes the sine ratio (opposite & hypotenuse).
    C only works for the acute angles.
    D only uses the reciprocals.
    Answer:

    The correct statement is choice .

  2. The table below shows (proves) that if we use the sine rule with a 90° angle it simplifies into the sine ratio for right-angled triangles. However, there is one missing step and one missing reason. Complete the proof by choosing the correct options from the lists available.

    Answer:
    Steps Reasons
    Given
    sinXx=sinZz Sine rule for ΔXYZ
    sinXx=sin(90°)z
    sinXx=1z Evaluate sin90°=1
    Multiply by x
    sinX=oppositehypotenuse Definition of opposite & hypotenuse


ID is: 3500 Seed is: 8910

The flexibility of the sine rule

The diagram below shows a triangle, drawn to scale. Two angles are shown, with values of 110° and 32°. The side of the triangle across from the 32 angle has a length of 6,1 units.

  1. Suppose you need to calculate x, the side across from the 110 angle. The two equations below are both accurate and you can use either of these versions to find x. But one choice makes the solution a bit easier. Which version is the better choice?

    Version 1 sin(110°)x=sin(32°)6,1
    Version 2 xsin(110°)=6,1sin(32°)
    Answer: The better choice is: .
  2. Now solve for the missing side, x. Use the equation identified in Question 1.

    INSTRUCTION: Round your answer to the tenths place (one decimal) if necessary.
    Answer: The length of the missing side is units.
    numeric

ID is: 3500 Seed is: 6618

The flexibility of the sine rule

The diagram below shows a triangle, drawn to scale. Two angles are shown, with values of 87° and 38°. The side of the triangle across from the 87 angle has a length of 13,5 units.

  1. Suppose you need to calculate x, the side across from the 38 angle. The two equations below are both accurate and you can use either of these versions to find x. But one choice makes the solution a bit easier. Which version is the better choice?

    Version 1 sin(38°)x=sin(87°)13,5
    Version 2 xsin(38°)=13,5sin(87°)
    Answer: The better choice is: .
  2. Now solve for the missing side, x. Use the equation identified in Question 1.

    INSTRUCTION: Round your answer to the tenths place (one decimal) if necessary.
    Answer: The length of the missing side is units.
    numeric


ID is: 3556 Seed is: 8123

Working algebraically: finding a sine ratio

Triangle DEF is shown below with sides labelled 12y4, 4y, and 3y4. All three of these expression are non-zero. E is labelled β. The diagram is not drawn to scale.

  1. Determine an expression for sinβ in terms of y and sinD.

    Answer: sinβ=
    expression
  2. Question 1 states that the three side expressions are non-zero. Why must those expressions be non-zero?

    Answer:

    The expressions must be non-zero .


ID is: 3556 Seed is: 5355

Working algebraically: finding a sine ratio

Triangle DEF is shown below with sides labelled 4b3, 4b, and 8b5. All three of these expression are non-zero. E is labelled θ. The diagram is not drawn to scale.

  1. Determine an expression for sinθ in terms of b and sinD.

    Answer: sinθ=
    expression
  2. Question 1 states that the three side expressions are non-zero. Why must those expressions be non-zero?

    Answer:

    The expressions must be non-zero .

7. The cosine rule



ID is: 4394 Seed is: 2788

Application of cos rule to prove an identity

Adapted from DBE Nov 2016, Grade 11, P2, Q7
Maths formulas

Given the cos rule, a2=b2+c22bccosA, Amirah was asked to prove that:

1+cosA = (b+c+a)(b+ca)2bc

Amirah has already answered the question below. But, she has made a mistake. Here is Amirah's answer, labelled Line 1 to Line 8:

1:a2=b2+c22bccosA2:2bccosA=b2+c2a23:cosA=b2+c2a22bc4:1+cosA=1+b2+c2a22bc5:1+cosA=2bc+b2+c2a22bc6:1+cosA=[(b2+2bc+c2)(a)2]2bc7:1+cosA=[(b+c)2(a)2][÷2bc]8:1+cosA=(b+c+a)(b+ca)2bc
Answer:
  1. The first line containing an error is .
  2. The correct replacement for this line is:

ID is: 4394 Seed is: 2159

Application of cos rule to prove an identity

Adapted from DBE Nov 2016, Grade 11, P2, Q7
Maths formulas

Given the cos rule, c2=a2+b22abcosC, Azubuike was asked to prove that:

cosC1 = (ab+c)(abc)2ab

Azubuike has already answered the question below. But, he has made a mistake. Here is Azubuike's answer, labelled Line 1 to Line 8:

1:c2=a2+b22abcosC2:2abcosC=a2+b2c23:cosC=a2+b2c22ab4:cosC1=a2+b2c22ab15:cosC1=a2+b2c22ab2ab6:cosC1=(a22ab+b2)(c)22ab7:cosC1=(a+b)2(c)22ab8:cosC1=(a+b+c)(a+bc)2ab
Answer:
  1. The first line containing an error is .
  2. The correct replacement for this line is:


ID is: 3503 Seed is: 6470

Finding an angle with the cosine rule

The triangle below has all three sides labelled. PR¯=7 units, RQ¯=4 units, and PQ¯=8 units. The figure is drawn to scale.

Find the measure of the angle R^.

TIP: The question states that the figure is drawn to scale. And the angle that you want looks like it is close to 90°. So make sure your answer agrees with that!
INSTRUCTION: Round your answer to two decimal places if necessary.
Answer:

The measure of angle R^ is °.

numeric

ID is: 3503 Seed is: 6719

Finding an angle with the cosine rule

The triangle below has all three sides labelled. CA¯=9 units, AB¯=11 units, and CB¯=10 units. The figure is drawn to scale.

Determine the measure of the angle B^.

TIP: The question states that the figure is drawn to scale. And the angle that you want is acute. In fact, it must be quite a bit less than 90°. So make sure your answer agrees with that!
INSTRUCTION: Round your answer to two decimal places if necessary.
Answer:

The measure of angle B^ is °.

numeric


ID is: 3505 Seed is: 9176

Finding the side opposite a known angle

The triangle below has two sides given: QP¯=8 and QR¯=9. The angle between these two sides is Q^=80°. These values are labelled in the diagram. The figure is drawn to scale.

Compute the length of the third side of the triangle, RP¯.

TIP: The question states that the figure is drawn to scale. The third side is clearly longer than either of the given sides. Therefore your answer should be greater than 9. If your answer does not agree with this, it probably means there is a mistake in your work.
INSTRUCTION: Round your answer to two decimal places if necessary.
Answer:

The length of side RP¯ is units.

numeric

ID is: 3505 Seed is: 1925

Finding the side opposite a known angle

The triangle below has two sides given: ZX¯=7 and ZY¯=10. The angle between these two sides is Z^=53°. These values are labelled in the diagram. The figure is drawn to scale.

Calculate the length of the third side of the triangle, YX¯.

TIP: The question states that the figure is drawn to scale. The third side is similar in length to the side with length 7. Therefore your answer should be somewhat close to 7. If your answer does not agree with this, it probably means there is a mistake in your work.
INSTRUCTION: Round your answer to two decimal places if necessary.
Answer:

The length of side YX¯ is units.

numeric


ID is: 3567 Seed is: 5714

Area calculations

A triangle has side lengths of 51,45 m, 54,2 m, and 64,82 m. Calculate the area of the triangle. Then select the correct units for the answer from the list of choices.

INSTRUCTION: Round the area to the hundredths place (two decimal places).
Answer:

The area of the triangle is .

one-of
type(numeric.abserror(0.004))

ID is: 3567 Seed is: 2655

Area calculations

A triangle has side lengths of 43,68 cm, 50,46 cm, and 74,86 cm. Calculate the area of the triangle. Then select the correct units for the answer from the list of choices.

INSTRUCTION: Round the area to the hundredths place (two decimal places).
Answer:

The area of the triangle is .

one-of
type(numeric.abserror(0.004))


ID is: 3591 Seed is: 4822

Cosine rule connections

  1. The cosine rule is about triangles, including right-angled triangles. And if you use a 90° angle in the cosine rule, it becomes the same as another important equation. Which statement below identifies what happens if we use a 90° angle in the cosine rule?

    Choices If you use a 90° angle is the cosine rule...
    A it becomes the theorem of Pythagoras
    B it changes into the sine rule
    C it becomes the equilateral triangle calculation
    D the sum of the interior angles of a triangle is 180°
    Answer: The correct statement is choice .
  2. The table below contains a proof. The proof shows that the cosine rule simplifies to the theorem of Pythagoras if the angle in the cosine rule is 90°. However, there is one step and one reason missing. Select the correct option from each list of choices to complete the proof.

    Answer:
    Steps Reasons
    Given
    c2=a2+b22abcosC^ Cosine rule using C^
    Substitute C^=90°
    c2=a2+b22ab(0) Evaluate cos(90°)=0
    c2=a2+b2
    Theorem of Pythagoras applies to ΔABC c is the hypotenuse, a and b are legs

ID is: 3591 Seed is: 2388

Cosine rule connections

  1. The cosine rule is about triangles, including right-angled triangles. And if you use a 90° angle in the cosine rule, it becomes the same as another important equation. Which statement below identifies what happens if we use a 90° angle in the cosine rule?

    Choices If you use a 90° angle is the cosine rule...
    A the sum of the interior angles of a triangle is 180°
    B you get all the special triangle angle choices
    C it becomes the theorem of Pythagoras
    D it changes into the area rule
    Answer: The correct statement is choice .
  2. The table below contains a proof. The proof shows that the cosine rule simplifies to the theorem of Pythagoras if the angle in the cosine rule is 90°. However, there is one step and one reason missing. Select the correct option from each list of choices to complete the proof.

    Answer:
    Steps Reasons
    Given
    c2=a2+b22abcosC^
    c2=a2+b22abcos(90°) Substitute C^=90°
    c2=a2+b22ab(0) Evaluate cos(90°)=0
    Multiply by 0
    Theorem of Pythagoras applies to ΔABC c is the hypotenuse, a and b are legs


ID is: 4393 Seed is: 1570

Cos rule proof

Adapted from DBE Nov 2016, Grade 11, P2, Q7
Maths formulas

In the diagram below ΔABC is drawn with acute angle C^.

Prove that c2=a2+b22abcosC^.

  1. To prove c2=a2+b22abcosC^, we need to make a construction. Which of the following represents a correct construction for this proof?

    I II
    III IV

    Using the construction, select the correct expression for the length of CD and BD.

    TIP: On a piece of paper, draw the construction diagram so that you can work from it. Name the sides of the triangle in the construction using the lower case letter of the opposite angle, eg. Side a opposite A^.
    Answer:

    The correct construction is Option:

    CD =

    BD =

  2. Rethabile was asked to prove c2=a2+b22abcosC^.

    Using the correct construction, she labelled the length CD as x and wrote out the correct proof.

    TIP: On a piece of paper, draw this construction and write out the cos rule proof for yourself.

    Select the correct missing pieces so that Rethabile's proof is complete.

    Answer:
    Proof Steps
    1. =xa
    2. x=acosC^
    In ΔBAD
    3. c2=+(bx)2(Pythagoras)
    4. BD2=c2b2+2bxx2
    In ΔBDC
    5. a2=BD2+x2(Pythagoras)
    6. BD2=
    7. c2b2+2bxx2=a2x2
    8. c2=b2+a2+
    9. c2=a2+b22abcosC^

ID is: 4393 Seed is: 117

Cos rule proof

Adapted from DBE Nov 2016, Grade 11, P2, Q7
Maths formulas

In the diagram below ΔABC is drawn with acute angle C^.

Prove that c2=a2+b22abcosC^.

  1. To prove c2=a2+b22abcosC^, we need to make a construction. Which of the following represents a correct construction for this proof?

    I II
    III IV

    Using the construction, select the correct expression for the y-coordinate of B and the x-coordinate of D.

    TIP: On a piece of paper, draw the construction diagram so that you can work from it. Name the sides of the triangle in the construction using the lower case letter of the opposite angle, eg. Side a opposite A^.
    Answer:

    The correct construction is Option:

    y-coordinate of B =

    x-coordinate of D =

  2. Akeju was asked to prove c2=a2+b22abcosC^.

    Using the correct construction, she labelled the coordinates of A, B and C, and wrote out the correct proof.

    TIP: On a piece of paper, draw this construction and write out the cos rule proof for yourself.

    Select the correct missing pieces so that Akeju's proof is complete.

    Answer:
    Proof Steps
    1. A(b;0)B(;asinC^)C(0;0)
    2. AB2=AD2+BD2(Pythagoras)
    3. c2=(bacosC^)2+
    4. c2=b22abcosC^+a2cos2C^+
    5. c2=b22abcosC^+a2(cos2C^+sin2C^)
    6. c2=b22abcosC^+(1)
    7. c2=a2+b22abcosC^


ID is: 1555 Seed is: 5541

Using the cosine rule

Calculate the length of line segment RQ¯ in the triangle below. The triangle has sides QP¯ and RP¯ equal to 14 and 11,8 units respectively. Also P^=69°. The figure is drawn to scale.

TIP: The question states that the figure is drawn to scale. So if the unknown side looks longer than the other sides in the triangle, your answer must be larger that the sides shown. If it looks shorter, your answer must be smaller. And so on.
INSTRUCTION: Round your answer to two decimal places if necessary.
Answer: Side RQ¯ is units.
numeric

ID is: 1555 Seed is: 7797

Using the cosine rule

Calculate the length of line segment RP¯ in the triangle below. The triangle has sides PQ¯ and RQ¯ equal to 6,6 and 8,6 units respectively. Also Q^=113°. The figure is drawn to scale.

TIP: The question states that the figure is drawn to scale. So if the unknown side looks longer than the other sides in the triangle, your answer must be larger that the sides shown. If it looks shorter, your answer must be smaller. And so on.
INSTRUCTION: Round your answer to two decimal places if necessary.
Answer: Side RP¯ is units.
numeric


ID is: 3568 Seed is: 2929

Predator and prey

A bird just landed on the ground, and there are two crickets nearby. This is good news for the bird because it is hungry. The distances between the bird and the two crickets are 30,12 cm and 35,71 cm. The crickets are 50,16 cm away from each other.

  1. What is the angle subtended by the two crickets at the bird's position?

    INSTRUCTION: Round to three decimal places.
    Answer: The angle is °.
    numeric
  2. If you are now told that the angle subtended at one of the crickets by the other cricket and the bird is 44,7°, determine the size of the angle subtended at the other cricket by the other animals.

    INSTRUCTION: Round to three decimal places.
    Answer: The angle is °.
    one-of
    type(numeric.abserror(0.0004))

ID is: 3568 Seed is: 316

Predator and prey

In the corner of a room, a gecko and two flies are on the wall. The gecko is hungry and hoping for a meal. The flies are 72,51 cm away from each other. The gecko is 37,5 cm away from one of the flies and 62,55 cm away from the other fly.

  1. What is the angle subtended by the two flies at the gecko's position?

    INSTRUCTION: Round to three decimal places.
    Answer: The angle is °.
    numeric
  2. If you are now told that the angle subtended at one of the flies by the other fly and the gecko is 59,6°, determine the size of the angle subtended at the other fly by the other animals.

    INSTRUCTION: Round to three decimal places.
    Answer: The angle is °.
    one-of
    type(numeric.abserror(0.0004))


ID is: 3584 Seed is: 8805

Perimeter problems

Triangle KLM has one angle and two sides known: L^=86,4° is the angle between the sides with lengths 11,7 cm and 18,6 cm. The figure is not drawn to scale.

Determine the perimeter of the triangle. Then select the correct units to go with the answer.

INSTRUCTION: Round your answer to two decimal places.
Answer:

The perimeter is .

numeric

ID is: 3584 Seed is: 8719

Perimeter problems

Triangle KML has one angle and two sides known: M^=79,8° is the angle between the sides with lengths 10 mm and 12,7 mm. The figure is not drawn to scale.

Calculate the perimeter of the triangle. Then select the correct units to go with the answer.

INSTRUCTION: Round your answer to two decimal places.
Answer:

The perimeter is .

numeric


ID is: 3587 Seed is: 5030

Isosceles triangles

Imagine an isosceles triangle. This triangle has one side with a length of 2,8 units. The length of the two congruent (equal) sides is unknown. However, the two congruent angles are both equal to 52,8°.

  1. What is the length of each of the two congruent sides?

    INSTRUCTION: Round your answer to two decimal places.
    Answer: Each of the congruent sides is units long.
    numeric
  2. Imagine that the two congruent sides of the triangle double in length while the third side remains unchanged. As a result, what will happen to the 52,8° angles?

    Answer: The 52,8° angles will .

ID is: 3587 Seed is: 6271

Isosceles triangles

Imagine an isosceles triangle. This triangle has one side with a length of 3,7 units. The length of the two congruent (equal) sides is unknown. However, the two congruent angles are both equal to 56°.

  1. What is the length of each of the two congruent sides?

    INSTRUCTION: Round your answer to two decimal places.
    Answer: Each of the congruent sides is units long.
    numeric
  2. Imagine that the two congruent sides of the triangle double in length while the third side remains unchanged. As a result, what will happen to the 56° angles?

    Answer: The 56° angles will .


ID is: 3589 Seed is: 3296

Figuring out what information to use

The figure below shows triangle BDA and triangle BCA, which have a common side BA¯. In triangle BCA we know side BC¯=4,5 and angle BCA=72°. In triangle BDA, we know sides BA¯=4,7 and DB¯=4,1, and we know BDA=76°. AD¯ is unknown and labelled '?'.

Answer the two questions below about this diagram and the cosine rule.

Answer:
  1. Can you find the missing side with the cosine rule?
  2. How do you know? .

ID is: 3589 Seed is: 8340

Figuring out what information to use

The figure below shows triangle CDB and triangle CAB, which have a common side CB¯. In triangle CDB, we know sides CB¯=4,3 and DC¯=3,8, and we know CDB=70°. In triangle CAB we know side AB¯=2,3 and angle CAB=80°. BD¯ is unknown and labelled '?'.

Answer the two questions below about this diagram and the cosine rule.

Answer:
  1. Can you find the missing side with the cosine rule?
  2. How do you know? .


ID is: 3611 Seed is: 9138

Finding an adjacent side

The diagram below is roughly to scale. It shows triangle ZXY. Side ZX is 16 units long and side ZY is 12 units long. Y^=θ. The value of cosθ=12. Determine the length of the third side, XY.

TIP: The question states that the figure is drawn roughly to scale. Make sure your answer is reasonable compared to the appearance of the figure.
INSTRUCTION: Round your answer to two decimal places if necessary.
Answer: Side XY¯ is units.
numeric

ID is: 3611 Seed is: 6514

Finding an adjacent side

The diagram below is roughly to scale. It shows triangle ZXY. Side ZX is 17 units long and side ZY is 14 units long. Y^=θ. The value of cosθ=314. Determine the length of the third side, XY.

TIP: The question states that the figure is drawn roughly to scale. Make sure your answer is reasonable compared to the appearance of the figure.
INSTRUCTION: Round your answer to two decimal places if necessary.
Answer: Side XY¯ is units.
numeric


ID is: 1432 Seed is: 6461

The cosine rule: identifying the correct sides and angles

The triangle below has sides labelled m,p, and n, and angles labelled θ,β, and α.

The cosine rule states:

(?)2=n2+p22npcosθ

Which of the following choices belongs in the place of the '?' in the formula?

Answer: The missing quantity is: .

ID is: 1432 Seed is: 8302

The cosine rule: identifying the correct sides and angles

The triangle below has sides labelled x,z, and y, and angles labelled X,Z, and Y.

The cosine rule states:

z2=y2+(?)22y(?)cosZ

Which of the following choices belongs in the place of the '?' in the formula?

Answer: The missing quantity is: .


ID is: 3552 Seed is: 568

The cosine rule: finding mistakes

The triangle below has sides 5, 6, and 7, and one angle labelled as A. A friend of yours writes down the cosine rule for this triangle, but there is a mistake.

Incorrect equation:

52=72+622(7)(6)cosA

How can your friend correct the equation?

Answer:

To fix the equation, your friend should .


ID is: 3552 Seed is: 7186

The cosine rule: finding mistakes

The triangle below has sides 4, 5, and 6, and one angle labelled as C. A friend of yours writes down the cosine rule for this triangle, but there is a mistake.

Incorrect equation:

42=52+622(5)(6)cosC

How can your friend correct the equation?

Answer:

To fix the equation, your friend should .



ID is: 3586 Seed is: 7209

Word problems: triangles

  1. In triangle QPR, side QP is 7 cm long. Side PR of the triangle is 2 cm more than side QR. If cosQ^=114, determine the lengths of the two unknown sides of the triangle.

    INSTRUCTION:
    1. The order of your answers does not matter.
    2. Your answer should be exact (do not round off).
    Answer:

    The lengths of the unknown sides are cm and cm.

    numeric
    numeric
  2. If Q^ is doubled, the opposite side, PR, will get bigger. However, side PR will not double. What does this mean about the relationship between angle Q^ and the length of side PR?

    Answer: It means that .

ID is: 3586 Seed is: 6655

Word problems: triangles

  1. Side QR of triangle PQR is 4 m more than side PR. Side PQ is 7 m long. Given that cosP^=514, what are the lengths of the two unknown sides of the triangle?

    INSTRUCTION:
    1. The order of your answers does not matter.
    2. Your answer should be exact (do not round off).
    Answer:

    The lengths of the unknown sides are m and m.

    numeric
    numeric
  2. If P^ is reduced by half, the opposite side, QR, will get smaller. However, side QR will not reduce by half. What does this mean about the relationship between angle P^ and the length of side QR?

    Answer: It means that .


ID is: 3590 Seed is: 8989

Using the cosine rule: is there enough information?

The triangle below has two sides with lengths 4 and 6. The triangle has two known angles, which are 58° and 86°. One of the sides is missing and it is labelled '?'.

Answer the two questions below about this diagram and the cosine rule.

Answer:
  1. Can you find the missing side using the cosine rule?
  2. Why / why not?

ID is: 3590 Seed is: 5749

Using the cosine rule: is there enough information?

The triangle below has a side with a length of 8. One of the angles is 48° and another angle is 38°. One of the angles is missing and it is labelled '?'.

Answer the two questions below about this diagram and the cosine rule.

Answer:
  1. Can you find the missing angle using the cosine rule?
  2. Why / why not?

8. Two-dimensional problems



ID is: 3566 Seed is: 7385

Algebra and the cosine rule

Adapted from DBE Nov 2015, Grade 11, P2, Q7
Maths formulas

In the diagram, not drawn to scale, RQ is the diameter of the circle. Triangle RPQ is drawn with vertex P outside the circle. Q^=x, RQ=4d, PQ=8d, and RP=5d.

  1. Determine the value of cosx.

    INSTRUCTION: Give your answer as a simplified fraction.
    Answer:

    cosx=

    numeric
  2. PQ cuts the circumference of the circle at T.

    1. Determine the size of angle RT^Q and select the correct reason.
    2. Determine RT in terms of x and d.
    Answer:
    1. RT^Q= °
    2. RT=
    numeric

ID is: 3566 Seed is: 1025

Algebra and the cosine rule

Adapted from DBE Nov 2015, Grade 11, P2, Q7
Maths formulas

In the diagram, not drawn to scale, QP is the diameter of the circle. Triangle QRP is drawn with vertex R outside the circle. P^=x, QP=5y, RP=5y, and QR=4y.

  1. Determine the value of cosx.

    INSTRUCTION: Give your answer as a simplified fraction.
    Answer:

    cosx=

    numeric
  2. RP cuts the circumference of the circle at T.

    1. Determine the size of angle QT^P and select the correct reason.
    2. Determine QT in terms of x and y.
    Answer:
    1. QT^P= °
    2. QT=
    numeric


ID is: 4392 Seed is: 1481

Using the sin rule & cos rule to find unknown sides & angles

Adapted from DBE Nov 2016, Grade 11, P2, Q7
Maths formulas

Quadrilateral JKLM is drawn with KL=105 mm and JK=132 mm. It is also given that JM^K=91,50°, MJ^K=58,25°, and LK^M=64,04°.

  1. Determine the length of KM.
  2. Determine the length of LM.
INSTRUCTION: Round your answer to two decimal places, if necessary.
Answer:
  1. KM= mm
  2. LM= mm
one-of
type(numeric.abserror(0.01))
one-of
type(numeric.abserror(0.01))

ID is: 4392 Seed is: 7056

Using the sin rule & cos rule to find unknown sides & angles

Adapted from DBE Nov 2016, Grade 11, P2, Q7
Maths formulas

Quadrilateral ABCD is drawn with BC=90 cm and CD=88 cm. It is also given that AD^B=92,62°, DA^B=41,23°, and BC^D=108,57°.

  1. Determine the length of BD.
  2. Determine the length of AB.
INSTRUCTION: Round your answer to two decimal places, if necessary.
Answer:
  1. BD= cm
  2. AB= cm
one-of
type(numeric.abserror(0.01))
one-of
type(numeric.abserror(0.01))